Im
jk ω0t
e
3
kω0t
Re
There was a celebrated Fourier at the Academy of Science,
whom posterity has forgotten; and in some garret an
obscure Fourier, whom the future will recall.
Victor Hugo Les Misèrables
Fourier Analysis and Synthesis
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
Introduction
Fourier Series
Fourier Transform FFT
Discrete Fourier Transform
Short Time Fourier Transform
Fast Fourier Transform
2-D Discrete Fourier Transform
Discrete Cosine Transfrom
J
3.9 Applications
Spectrogram
Based Digital Filters
Digital BASS for Audio
Radar Signal Processing
Pitch Modification
Image Compression
Watermarking
Bibliography, Exercises
ean Baptiste Joseph Fourier (1768-1830) studied the mathematical theory of
heat conduction in his major work, The Analytic Theory of Heat, (Théorie
analytique de la chaleur). He established the partial differential equation
governing heat diffusion and solved it using an infinite series of trigonometric
functions. The description of a signal in terms of elementary trigonometric
functions had a profound effect on the way signals are analysed. The Fourier
method is the most extensively applied signal-processing tool. This is because the
transform output lends itself to easy interpretation and manipulation, and leads to
the concept of frequency analysis. Furthermore even biological systems such as
the human auditory system perform some form of frequency analysis of the input
signals. This chapter begins with an introduction to the complex Fourier series and
the Fourier transform, and then considers the discrete Fourier transform, the Fast
Fourier transform, the 2-D Fourier transform and the discrete cosine transform.
Important engineering issues such as the trade-off between the time and frequency
resolutions, problems with finite data length, windowing and spectral leakage are
considered. The applications of the Fourier transform include filtering.
telecommunication, music processing, pitch modification, signal coding and signal
synthesis feature extraction for pattern identification as in speech recognition,
image processing, spectral analysis in astrophysics, radar signal processing.
2
Chap.3 Fourier Analysis and Synthesis
3.1 Introduction
The objective of signal transformation is to express a signal as a combination of a
set of basic “building block” signals, known as the basis functions. The transform
output should lend itself to convenient analysis, interpretation and manipulation.
A useful consequence of transforms, such as the Fourier and the Laplace, is that
differential analysis on the time domain signal become simple algebraic operations
on the transformed signal. In the Fourier transform the basic building block
signals are sinusoidal signals with different periods giving rise to the concept of
frequency. In Fourier analysis a signal is decomposed into its constituent
sinusoids, i.e. frequencies, the amplitudes of various frequencies form the socalled frequency spectrum of the signal. In an inverse Fourier transform operation
the signal can be synthesised by adding up its constituent frequencies. It turns out
that many signals that we encounter in daily life such as speech, car engine noise,
bird songs, music etc. have a periodic or quasi-periodic structure, and that the
cochlea in the human hearing system performs a kind of harmonic analysis of the
input audio signals. Therefore the concept of frequency is not a purely
mathematical abstraction in that biological and physical systems have also evolved
to make use of the frequency analysis concept.
The power of the Fourier transform in signal analysis and pattern recognition is its
ability to reveals spectral structures that may be used to characterise a signal. This
is illustrated in Fig. 3.1 for the two extreme cases of a sine wave and a purely
random signal. For a periodic signal the power is concentrated in extremely
narrow bands of frequencies indicating the existence of structure and the
predictable character of the signal. In the case of a pure sine wave as shown in Fig.
3.1.a the signal power is concentrated in one frequency. For a purely random
signal as shown in Fig 3.1.b the signal power is spread equally in the frequency
domain indicating the lack of structure in the signal.
P XX (f)
x(t)
t
(a)
P XX (f)
x(t)
t
(b)
Figure 3.1 The concentration or spread of power in frequency indicates the correlated or
random character of a signal: (a) a predictable signal, (b) a random signal.
Sec. 3.2 Fourier Series
3
Notation: In this chapter the symbols t and m denote continuous and discrete
time variables, and f and k denote continuous and discrete frequency variables
respectively. The variable ω=2πf denotes the angular frequency in units of rad/s
and is used interchangably (within a scaling of factor of 2π) with the frequency
variable f in units of Hz.
3.2 Fourier Series: Representation of Periodic Signals
The following three sinusoidal functions form the basis functions for the Fourier
analysis
x1 (t ) = cosω 0 t
(3.1)
x 2 (t ) = sinω 0 t
(3.2)
(3.3)
x3 (t ) = cosω 0 t + j sinω 0 t = e jω0t
Fig. 3.2.a shows the cosine and the sine components of the complex exponential
(cisoidal) signal of Eq. (3.3), and Fig. 3.2.b shows a vector representation of the
complex exponential in a complex plane with real (Re) and imaginary (Im)
dimensions. The Fourier basis functions are periodic with an angular frequency of
ω0 rad/s and a period of T0=2π/ω0=1/F0 seconds, where F0 is the frequency in
Hz. The following properties make the sinusoids the ideal choice as the
elementary building block basis functions for signal analysis and synthesis:
(i) Orthogonality; two sinusoidal functions of different frequencies have the
following orthogonal property :
Im
sin(kωot)
jk ω0t
e
cos(kωot)
kω0t
Re
t
T0
(a)
(b)
Figure 3.2 - Fourier basis functions: (a) real and imaginary parts of a complex sinusoid,
(b) vector representation of a complex exponential.
4
Chap.3 Fourier Analysis and Synthesis
∞
∞
∞
1
1
∫−∞sin(ω1t ) sin(ω 2 t ) dt = − 2 −∫∞cos(ω1 + ω 2 )t dt + 2 −∫∞cos(ω1 − ω 2 )t dt = 0
(3.4)
For harmonically related sinusoids the integration can be taken over one period.
Similar equations can be derived for the product of cosines, or sine and cosine,
of different frequencies. Orthogonality implies that the sinusoidal basis
functions are independent and can be processed independently. For example in
a graphic equaliser we can change the relative amplitudes of one set of
frequencies, such as the bass, without affecting other frequencies, and in
subband coding different frequency bands are coded independently and
allocated different number of bits.
(ii) Sinusoidal functions are infinitely differentiable. This is important, as most
signal analysis, synthesis and manipulation methods require the signals to be
differentiable.
(iii) Sine and cosine signals of the same frequency have only a phase difference of
π/2 or equivalently a relative time delay of a quarter of one period i.e. T0/4.
Associated with the complex exponential function e jω 0 t is a set of harmonically
related complex exponential of the form
[1, e
± jω 0 t
,e ± j2ω 0t ,e ± j3ω 0t , K
]
(3.5)
The set of exponential signals in Eq. (3.5) are periodic with a fundamental
frequency ω0=2π/T0=2πF0 where T0 is the period and F0 is the fundamental
frequency. These signals form the set of basis functions for the Fourier analysis.
Any linear combination of these signals of the form
∞
∑c e
k
jkω 0t
(3.6)
k = −∞
is also periodic with a period of T0. Conversely any periodic signal x(t) can be
synthesised from a linear combination of harmonically related exponentials.
The Fourier series representation of a periodic signal are given by the following
synthesis and analysis equations:
x(t ) =
∞
∑c e
k
k = −∞
jkω 0t
k = L − 1,0,1,L
Synthesis equation (3.7)
Sec. 3.2 Fourier Series
1
ck =
T0
T0 / 2
5
∫ x(t )e
− jkω 0t
dt
k = L − 1,0,1,L
Analysis equation
(3.8)
−T0 / 2
The complex-valued coefficient ck conveys the amplitude (a measure of the
strength) and the phase of the frequency content of the signal at kω0 Hz. Note from
the analysis Eq. (3.8), that the coefficient ck may be interpreted as a measure of
− jkω t
the correlation of the signal x(t) and the complex exponential e 0 .
The set of complex coefficients … c−1, c0, c1, … are known as the signal spectrum.
Eq. (3.7) is referred to as the synthesis equation, and can be used as a frequency
synthesizer (as in music synthesizers) to generate a signal as a weighted
combination of its elementary frequencies. The representation of a signal in the
form of Eq. (3.7) as the sum of its constituent harmonics is also referred to as the
complex Fourier series representation. Note from Eqs. (3.7) and (3.8) that the
complex exponentials that form a periodic signal occur only at discrete frequencies
which are integer multiples, i.e. harmonics, of the fundamental frequency ω0.
Therefore the spectrum of a periodic signal, with a period of T0, is discrete in
frequency with discrete spectral lines spaced at integer multiples of ω0=2π/T0.
Example 3.1 Given the Fourier synthesis Eq. (3.7), obtain the frequency analysis
Eq. (3.8).
Solution: Multiply both sides of Eq. (3.7) by e − jmω0t and integrate over one period
to obtain
T0 / 2
∫ x(t )e
−T0 / 2
− jmω 0t
dt =
T0 / 2
∞
∑ c ∫e
k
k = −∞
jkω 0t
e
− jmω 0t
dt =
∑ c ∫e
k
k = −∞
−T0 / 2
T0 / 2
∞
j( k − m )ω 0t
dt
(3.9)
−T0 / 2
From the orthogonality principle the integral in the r.h.s of Eq. (3.9) is zero unless
k=m in which case the integral is equal to T0. Hence
cm =
1
T0
T0 / 2
∫ x(t )e
−T0 / 2
− jmω 0t
dt
(3.10)
6
Chap.3 Fourier Analysis and Synthesis
| ck |
x(t)
t
-1
1
k
-103
103
f H
T0=1 ms
(a)
(b)
Figure 3.3 - A sinewave and its magnitude spectrum.
Example 3.2 Find the frequency spectrum of a 1 kHz sinewave shown in Fig
3.3.a
x(t ) = sin( 2000πt ) − ∞<t <∞
(3.11)
Solution A: The Fourier synthesis Eq. (3.7) can be written as
x(t ) =
∞
∑c e
k
jk 2000πt
k = −∞
= L + c −1e − j2000πt + c0 + c1e j2000πt + L
now the sine wave can be expressed as
1
1
x(t ) = sin(2000πt ) = e j2000πt − e − j2000πt
2j
2j
(3.12)
(3.13)
Equating the coefficients of Eqs. (3.12) and (3.13) yields
c1 =
1
1
, c −1 = −
and c k ≠ ±1 = 0
2j
2j
( 3.14)
Fig 3.3.b shows the magnitude spectrum of the sinewave, where the spectral lines
c1 and c−1 correspond to the 1 kHz and −1 kHz frequencies respectively.
Solution B: Substituting sin( 2000πt ) =
1 j2000πt 1 − j2000πt
in the Fourier
e
− e
2j
2j
analysis Eq. (3.8) yields
1
ck =
T0
=
T0 / 2
 1 j 2000πt 1 − j2000πt  − jk 2000πt
 e
 e
dt
− e
2j
2j

−T0 / 2 
1
2 jT0
∫
T0 / 2
∫
e j (1− k ) 2000πt dt −
−T0 / 2
1
2 jT0
T0 / 2
∫e
−T0 / 2
− j(1+ k ) 2000πt
dt
(3.15)
Sec. 3.2 Fourier Series
7
Since sine and cosine functions are positive-valued over one half a period and
negative-valued over the other half, it follows that Eq. (3.15) is zero unless k=1 or
k=−1.
1
1
c1 =
and c −1 = −
and c k ≠ ±1 = 0
(3.16)
2j
2j
Example 3.3 Find the frequency spectrum of a periodic train of pulses with
amplitude of 1.0, a period of 1.o kHz and a pulse 'on' duration of 0.3 milliseconds.
Solution: The pulse period T0=1/F0=0.001 s, and the angular frequency
ω0=2πF0=2000π rad/s. Substituting the pulse signal in the Fourier analysis Eq.
(3.8) gives
1
ck =
T0
T0 / 2
∫
x(t ) e − jkω 0t dt =
−T0 / 2
e − jk 2000πt
=
− j2πk
t = 0.00015
t = −0.00015
0.00015
1
e − jk 2000πt dt
0.001 −0.00015
∫
(3.17)
e j0.3πk − e − j0.3πk sin(0.3πk )
=
=
j2πk
πk
For k=0 as c0=sin(0)/0 is undefined, differentiate the numerator and denominator
of Eq. (3.17) w.r.t. to the variable k (strictly this can only be done for a continuous
variable k i.e. when the period T0 tends to infinity) to obtain
c0 =
0.3π cos(0.3π 0)
π
(3.18
= 0.3
x(t)
t
c(k)
0 .3
0 .2 5
0 .2
0 .1 5
0 .1
0 .0 5
0
-0 .0 5
1
2
k
-0 .1
Figure 3.4 - A rectangular pulse train and its discrete frequency ‘line’ spectrum.
8
Chap.3 Fourier Analysis and Synthesis
Example 3.4 For the example 3.3 write the formula for synthesising the signal up
to the Nth harmonic, and plot a few examples for the increasing number of
harmonics.
Solution: The equation for the synthesis of a signal upto the Nth harmonic content
is given by
x(t ) =
N
∑c
ke
jkω 0 t
= c0 +
k =− N
= c0 +
N
∑c e
k
jkω 0 t
+
k =1
N
∑ [Re(c
N
∑c
−k e
− jkω 0 t
k =1
k
) + j Im(c k )][cos(kω 0 t ) + j sin( kω 0 t )]
k
) − j Im(c k )][cos(kω 0 t ) − j sin(kω 0 t )]
k =1
+
N
∑ [Re(c
k =1
= c0 +
N
∑ [2 Re(c
k
(3.19)
) cos(kω 0 t ) − 2 Im(c k ) sin( kω 0 t )]
k =1
The following MatLab code generates a synthesised pulse train composed of N
harmonics. In this example there are 5 cycles in an array of 1000 samples. Fig. 3.5
shows the waveform for the number of harmonics equal to; 1,3,6, and 100.
NHarmonics=100; Ncycles=5; Nsamples=1000;
y(1:Nsamples)=0.3; j=1:Nsamples;
for k=1:NHarmonics
x(j)=(2*sin(0.3*pi*k)/(pi*k))*cos(k*2*pi*Ncycles*j/Nsamples);
y=y+x;
plot(y); pause;
end
x(t)
x(t)
(a)
(b)
t
t
x(t)
x(t)
(c)
(d)
t
t
Figure 3.5 Illustration of the Fourier synthesis of a periodic pulse train, and the Gibbs
phenomenon, with the increasing number of harmonics in the Fourier synthesis:
(a) N=1, (b) N=3, (c) N=6, and (d) N=100.
Sec. 3.3 Fourier Transform
9
3.2.1 Fourier Synthesis of Discontinuous Signals:
Gibbs Phenomenon
The sinusoidal basis functions of the Fourier transform are smooth and infinitely
differentiable. In the vicinity of a discontinuity the Fourier synthesis of a signal
exhibits ripples as shown in the Fig 3.5. The peak amplitude of the ripples does
not decrease as the number of harmonics used in the signal synthesis increases.
This behaviour is known as the Gibbs phenomenon. For a discontinuity of unity
height, the partial sum of the harmonics exhibits a maximum value of 1.09 (that is
an overshoot of 9%) irrespective of the number of harmonics used in the Fourier
series. As the number of harmonics used in the signal synthesis increases, the
ripples become compressed toward the discontinuity but the peak amplitude of the
ripples remains constant.
3.3 Fourier Transform: Representation of Aperiodic Signals
The Fourier series representation of periodic signals consist of harmonically
related spectral lines spaced at the integer multiples of the fundamental frequency.
The Fourier representation of aperiodic signals can be developed by regarding an
aperiodic signal as a special case of a periodic signal with an infinite period. If the
period of a signal is infinite, then the signal does not repeat itself and is aperiodic.
Now consider the discrete spectra of a periodic signal with a period of T0, as
c(k)
x(t)
Ton
(a)
Toff
t
T0=Ton+Toff
1
T0
k
X(f)
x(t)
Toff = ∞
(b)
t
f
Figure 3.6 – (a) A periodic pulse train and its line spectrum, (b) a single pulse from the
periodic train in (a) with an imagined ‘off’ duration of infinity; its spectrum is the envelope
of the spectrum of the periodic signal in (a).
10
Chap.3 Fourier Analysis and Synthesis
shown in Fig. 3.6.a. As the period T0 is increased, the fundamental
frequency F0=1/T0 decreases, and successive spectral lines become more closely
spaced. In the limit as the period tends to infinity (i.e. as the signal becomes
aperiodic) the discrete spectral lines merge and form a continuous spectrum.
Therefore the Fourier equations for an aperiodic signal, (known as the Fourier
transform), must reflect the fact that the frequency spectrum of an aperiodic signal
is continuous. Hence to obtain the Fourier transform relation the discretefrequency variables and operations in the Fourier series Eqs. (3.7) and (3.8) should
be replaced by their continuous-frequency counterparts. That is the discrete
summation sign
Σ should be replaced by the continuous summation integral
∫
,
the discrete harmonics of the fundamental frequency kF0 should be replaced by the
continuous frequency variable f, and the discrete frequency spectrum ck must be
replaced by a continuous frequency spectrum say X ( f ) . The Fourier synthesis
and analysis equations for aperiodic signals, the so-called Fourier transform pair,
are given by
∞
x(t ) =
∫ X ( f )e
j2πft
df
(3.20)
X ( f ) = x(t )e − j2πft dt
(3.21)
−∞
∞
∫
−∞
Note from Eq. (3.21), that X ( f ) may be interpreted as a measure of the
− j2πft
.
correlation of the signal x(t) and the complex sinusoid e
The condition for existence and computability of the Fourier transform integral of
a signal x(t) is that the signal must have finite energy
∞
∫
2
x(t ) dt < ∞
(3.22)
−∞
Example 3.5 Derivation of inverse Fourier transform Given the Fourier
transform Eq. (3.21) derive the Fourier synthesis Eq. (3.20).
Solution: Consider the Fourier analysis Eq. (3.8) for a periodic signal and Eq.
(3.21) for its non-periodic version (consisting of one period only). Comparing
these equations reproduced below
ck =
we have
1
T0
T0 / 2
∫
x(t )e − j2πkF0 t dt
− T0 / 2
∞
∫
X ( f ) = x(t )e − j2πft dt
−∞
Sec. 3.3 Fourier Transform
11
ck =
1
X (kF0 )
T0
(3.23)
as T0 → ∞
where F0=1/T0. Using Eq. (3.23) the Fourier synthesis Eq. (3.7) for a periodic
signal can be rewritten as
x(t ) =
∞
∑ X ( k / T )e
0
j2πkF0 t
∆F
(3.24)
k = −∞
where ∆F=1/T0=F0 is the frequency spacing between successive spectral lines of
the spectrum of a periodic signal as shown in Fig. 3.6. Now as the period T0 tends
to infinity, ∆F=1/T0 tends to zero, then the discrete frequency variables kF0 and
∆F should be replaced by a continuous frequency variable f, and the discrete
summation sign by the continuous integral sign. Thus Eq. (3.24) becomes
∞
x(t ) ⇒
T0 →∞
∫ X ( f )e
j2πft
(3.25)
df
−∞
Example 3.6 The spectrum of an Impulse Function
Consider the unit-area pulse p(t) shown in Fig 3.7.a. As the pulse width ∆ tends to
zero the pulse tends to an impulse. The impulse function shown in Fig 3.7.b is
defined as a pulse with an infinitesimal time width as
1 / ∆ t ≤ ∆ / 2
 0 t > ∆ / 2
δ (t ) = limit p (t ) = 
∆→0
p(t)
1/∆
∆
(a)
(3.26)
δ(t)
As ∆
∆(f)
0
t
t
(b)
Figure 3.7 (a) A unit-area pulse, (b) the pulse becomes an impulse as
spectrum of the impulse function.
f
(c)
∆ → 0 , (c) the
12
Chap.3 Fourier Analysis and Synthesis
It is easy to see that the integral of the impulse function is given by
∞
1
∫ δ (t )dt = ∆ × ∆ =1
(3.27)
−∞
The important sampling property of the impulse function is defined as
∞
∫ x(t )δ (t − T ) dt = x(T )
(3.28)
−∞
The Fourier transform of the impulse function is obtained as
∞
∫
∆ ( f ) = δ (t )e − j 2πft dt = e 0 = 1
(3.29)
−∞
The impulse function is used as a test function to obtain the impulse response of a
system. This is because as shown in Fig 3.7.c an impulse is a spectrally rich signal
containing all frequencies in equal amounts.
Example 3.7 Find the spectrum of the 10 kHz periodic pulse train shown in Fig.
3.8.a with an amplitude of 1 mV and expressed as
x(t ) =
∞
∑
m = −∞
A× limit p(t − mT0 ) × ∆ =
∆ →0
∞
∑ A δ (t − mT )
(3.30)
0
m = −∞
where p(t) is a unit-area pulse with width ∆ as shown in Fig. 3.7.a, and the
function δ(t-mT0) is now assumed to have a unit amplitude representing the area
under the impulse.
Solution: T0=1/F0=0.1 ms and A=1 mV. For the time interval −T0/2<t<T0/2
x(t)=Aδ(t), hence we have
ck
x(t )
...
T0
...
t
1/T0
(a)
(b)
Figure 3.8 A periodic impulse train x(t) and its Fourier series spectrum ck.
k
Sec. 3.3 Fourier Transform
1
ck =
T0
T0 / 2
∫ x(t )e
− jkω 0t
−T0 / 2
13
1
dt =
T0
T0 / 2
∫ Aδ (t )e
− jkω 0t
dt =
−T0 / 2
A
= 10 V
T0
(3.31)
As shown in Fig. 3.8.b the spectrum of a periodic impulse train in time is a
periodic impulse train in frequency
Example 3.8 The Spectrum of a Rectangular Function: Sinc Function
The rectangular pulse is particularly important in digital signal analysis and digital
communication. For example, the spectrum of a rectangular pulse can be used to
calculate the bandwidth required by pulse radar systems or communication
systems that transmit pulses. The Fourier transform of a rectangular pulse Fig.
3.9.a of duration T seconds is obtained as
∞
T /2
−∞
−T / 2
R ( f ) = ∫ r (t ) e − j2πft dt =
∫ 1.e
− j2πft
dt
e j2πfT / 2 − e − j2πfT / 2
sin(πfT )
=
=T
= T sinc(πfT )
j2πf
πfT
(3.32)
Fig. 3.9.b shows the spectrum of the rectangular pulse. Note that most of the pulse
energy is concentrated in the main lobe within a bandwidth of 2/T. However there
are pulse energy in the side lobes that may interfere with other electronic devices
operating at the side lobe frequencies.
Matlab code for drawing the spectrum of a rectangular pulse.
% Pulsewidth T, Number of frequency samples N, Frequency resolution for plot df.
T=.001;df=10; N=1000;
for i=1:N
if (i~=N/2)x(i)=T*sin(2*pi*df*T*(i-N/2))/(2*pi*df*T*(i-N/2));end
end
x(N/2)=T; plot(x);
R(f)
r(t)
T
1
T
(a)
t
-2
T
-1
T
1
T
2
T
3
T
(b)
Figure 3.9 A rectangular pulse and its spectrum.
f
14
Chap.3 Fourier Analysis and Synthesis
e–σtIm[e–j2πft]
e–σtIm[e–j2πmf]
e–σtIm[e–j2πmf]
m
σ>0
m
m
σ =0
Figure 3.10 – The Laplace basis functions.
σ <0
3.3.1 The Relation Between the Laplace and the Fourier Transforms
The Laplace transform of x(t) is given by the integral
∞
X ( s ) = ∫ x(t )e − st dt
(3.33)
0−
where the complex variable s=σ+jω, and the lower limit of t=0− allows the
possibility that the signal x(t) may include an impulse. The inverse Laplace
transform is defined by
σ 1 + j∞
x(t ) =
∫ X (s)e
σ
st
ds
(3.34)
1 − j∞
where σ1 is selected so that X(s) is analytic (no singularities) for s>σ1. The basis
functions for the Laplace transform are damped or growing sinusoids of the form
e − st = e −σt e − jωt as shown in Fig. 3.10. These are particularly suitable for transient
signal analysis. The Fourier basis functions are steady complex exponential, e − jωt ,
of time-invariant amplitudes and phase, suitable for steady state or time-invariant
signal analysis.
The Laplace transform is a one-sided transform with the lower limit of integration
at t = 0 − , whereas the Fourier transform Eq. (3.21) is a two-sided transform with
the lower limit of integration at t = −∞ . However for a one-sided signal, which is
zero-valued for t < 0 − , the limits of integration for the Laplace and the Fourier
transforms are identical. In that case if the variable s in the Laplace transform is
replaced with the frequency variable jω then the Laplace integral becomes the
Fourier integral. Hence for a one-sided signal, the Fourier transform is a special
case of the Laplace transform corresponding to s=jω and σ=0. The relation
between the Fourier and the Laplace transforms are discussed further in Chapter 4.
3.3.2 Properties of the Fourier Transform
Sec. 3.3 Fourier Transform
15
There are a number of Fourier Transform properties that provide further insight
into the transform and are useful in reducing the complexity of the solutions of
Fourier transforms and inverse transforms. These are:
Linearity
The Fourier transform is a linear operation, this mean the principle of
superposition applies. Hence
if:
z (t ) = ax(t )+by (t )
(3.35)
then
Z ( f ) = aX ( f )+bY ( f )
(3.36)
Symmetry
This property states that if the time domain signal x(t) is real (as is often the case
in practice) then
X ( f ) = X * (− f )
(3.37)
Where the superscript asterisk * denotes the complex conjugate operation. From
Eq. (3.37) it follows that Re{ X ( f ) } is an even function of f and Im{ X ( f ) } is an
odd function of f. Similarly the magnitude of X ( f ) is an even function and the
phase angle is an odd function.
Time Shifting and Frequency Modulation (FM)
Let X ( f ) = F [ x(t ) ] be the Fourier transform of x(t). If the time domain signal x(t)
is delayed by an amount T0, the effect on its spectrum X ( f ) is a phase shift of
e − j2πT0 f as
F [x(t − T0 )] = e − j2πfT
0
X(f )
(3.38)
Conversely if X(f) is shifted by an amount F0, the effect is
F −1 [X ( f
− F0 )] =e j2πF0t x(t )
(3.39)
Note that the modulation Eq. (3.39) states that multiplying a signal x(t) by e j 2πF0t
translates the spectrum of x(t) onto the frequency F0, this is the frequency
modulation (FM) principle.
Differentiation and Integration
Let x(t) be a continuous time signal with Fourier transform X ( f ) .
∞
x(t ) =
∫ X ( f )e
−∞
j2πft
df
(3.40)
16
Chap.3 Fourier Analysis and Synthesis
Then by differentiating both sides of the Fourier transform Eq. (3.40) we obtain
∞
d x(t )
= j2πf X ( f ) e j2πft df
14243
dt
−∞
∫
(3.41)
FourierTransform
of d x ( t ) / dt
That is multiplication of X ( f ) by the factor j2πf in the frequency domain is
equivalent to differentiation of x(t) in time. Similarly division of X ( f ) by j2πf is
equivalent to integration of the function of time x(t)
t
F
1
∫ x(τ )dτ ←→ j2πf X ( f )+πX (0)δ ( f )
(3.42)
−∞
Where the impulse term on the right-hand side reflects the dc or average value that
can result from the integration.
Time and Frequency Scaling
If x(t) and X(f) are Fourier transform pairs then
1 f
F
x(αt ) ←→
X 
α α 
(3.43)
For example try to say something very slowly, then α >1, your voice spectrum
will be compressed and you may sound like a slowed down tape or disc, you can
do the reverse and the spectrum would be expanded and your voice shifts to higher
frequencies, This property is further illustrated in section 3.9.4.
Convolution
The convolution integral of two signals x(t) and h(t) is defined as
∞
∫
y (t ) = x(τ ) h(t − τ )dτ
(3.44)
−∞
The convolution integral is also written as
y (t ) = x(t ) * h(t )
(3.45)
where asterisk * denotes the convolution operation. The convolution integral is
used to obtain the time-domain response of linear systems to arbitrary inputs as
will be discussed in later sections.
Sec. 3.3 Fourier Transform
17
The Convolution Property of the Fourier Transform. It can be shown that
convolution of two signals in the time domain corresponds to multiplication of the
signals in the frequency domain, and conversely multiplication in the time domain
corresponds to convolution in the frequency domain. To derive the convolutional
property of the Fourier transform take the Fourier transform of the convolution of
the signals x(t) and h(t) as
∞
∞
∞

 x(τ )h(t − τ )dτ  e − j2πft dt = x(τ ) e − j2πfτ dτ h(t − τ ) e − j2πf (t −τ ) dt


−∞  −∞
−∞
−∞

= X ( f )H ( f )
∞
∫ ∫
∫
∫
(3.46)
Duality
Comparing the Fourier transform and the inverse Fourier transform relations we
observe a symmetric relation between them. In fact the main difference between
Eqs. (3.20) and (3.21) is a negative sign in the exponent of e − j2πft in Eq (3.21).
This symmetry leads to a property of Fourier transform known as the duality
principle and stated as
F
x(t ) ←→ X ( f )
(3.47)
F
X (t ) ←→ x( f )
As illustrated in Fig 3.11. the Fourier transform of a rectangular function of time
r(t) has the form of a sinc pulse function of frequency sinc( f ) . From the duality
X (f)
x(t)
T
-2
T
t
T
-1
T
1
T
2
T
3
T
f
x(f)
X (t)
T
t
T1
-1
T1
1
T1
f
Figure 3.11 Illustration of the principle of duality.
principle the Fourier transform of a sinc function of time sinc(t) is a rectangular
function of frequency R(f).
18
Chap.3 Fourier Analysis and Synthesis
Parseval's Theorem: Energy Relationship in Time and Frequency
Parseval’s relation states that the energy of a signal can be computed by
integrating the squared magnitude of the signal either over the time domain or over
the frequency domain. If x(t) and X(f) are a Fourier transform pair, then
∞
Energy =
∫
∞
∫
| x(t ) | 2 dt = | X ( f ) | 2 df
−∞
(3.48)
−∞
This expression referred to as Parseval's relation follows from a direct application
of the Fourier transform.
Example 3.9 The Spectrum of a Finite Duration Signal
Find and sketch the frequency spectrum of the following finite duration signal
x(t ) = sin( 2πF0 t ) − NT0 / 2 ≤ t ≤ NT0 / 2
(3.49)
where T0=1/F0 is the period and ω0= 2π/T0.
Solution: Substitute for x(t) and its non-zero valued limits in the Fourier transform
Eq. (3.21)
NT0 / 2
X(f ) =
∫ sin(2πF t )e
0
− j 2πft
dt
− NT0 / 2
(3.50)
substituting sin(2πF0 t ) = (e j2πF0t − e − j2πF0t ) / 2 j in (3.50) gives
NT0 / 2
∫
X(f ) =
− NT0
e j2πF0t − e − j2πF0t − j2πft
e
dt
2
j
/2
NT0 / 2
=
∫
− NT0
NT0 / 2
e − j2π ( f − F0 )t
dt −
2j
/2
− NT
Evaluating the integrals yields
∫
0
e − j2π ( f + F0 )t
dt
2j
/2
(3.51)
Sec. 3.3 Fourier Transform
X(f ) =
=
19
e− jπ ( f − F0 ) NT0 − e jπ ( f − F0 ) NT0 e− jπ ( f + F0 ) NT0 − e jπ ( f + F0 ) NT0
−
4π ( f − F0 )
4π ( f + F0 )
j
−j
sin (π ( f − F0 ) NT0 ) +
sin (π ( f + F0 ) NT0 )
2π( f − F0 )
2π( f + F0 )
(3.52)
Matlab code for drawing the spectrum of a finite duration sinwave.
% Sinewave Period T0, Number of Cycles in the window N
T0=.001; F0=1/T0; N=2000;
for Nc=1:4;
for f=1:999
if ((f-f0)~=0)
x(Nc,f)=Nc*T0*sin(pi*(f-F0)*T0*Nc)/(pi*(f-F0)*T0*Nc); end
end
x(Nc,1000)=Nc*T0;
end plot(x);
X(f)
Nc=4
Nc=2
Nc=1
F0
f
Nc=3
Figure 3.12 The spectrum of a finite duration sine wave with the increasing length of
observation. Nc is the number of cycles in the observation window. Note that the
width of the main lobe depends inversely on the duration of the signal.
The signal spectrum can be expressed as the sum of two shifted sinc functions.
X ( f ) = − jNT0 sinc(π ( f − F0 ) NT0 ) + j NT0 sinc(π ( f + F0 ) NT0 )
(3.53)
Note that energy of a finite duration sinewave is spread in frequency between the
main lobe and the side lobes of the sinc function. Fig. 3.12 demonstrates that as
the window length increases the energy becomes more concentrated in the main
lobe and in the limit for an infinite duration window the spectrum tends to an
impulse positioned at the frequency F0.
20
Chap.3 Fourier Analysis and Synthesis
Example 3.10 Calculation of the bandwidth for transmission of data at a rate of
rb bits per second. In its simplest form binary data can be represented as a
sequence of amplitude modulated pulses as
N −1
x(t ) = ∑ A(m) p (t − mTb )
()
m =0
where A(m) may be +1 or a –1 and
1
p (t ) = 
0
t ≤ Tb / 2
()
t >0
The Fourier transform of x(t) is given by
N −1
N −1
m=0
m=0
X ( f ) = ∑ A(m) P( f )e − j 2πmTb f = P( f ) ∑ A(m)e − j 2πmTb f
()
The power spectrum of this signal is obtained as
∞
 ∞

E [X ( f ) X * ( f )] = E  ∑ A(m) P( f )e − j 2πmTb f ∑ A(n) P * ( f )e j 2πmTb f 
n = −∞
 m = −∞

= P( f )
2
N −1 N −1
∑ ∑ E[ A(m) A(n)]e
()
− j 2π ( m − n )Tb f
m =0 n =0
Now assuming that the data is uncorrelated we have
1
E[ A(m) A(n)] = 
0
m=n
m≠n
()
Substituting Equations () in () we have
E [ X ( f ) X * ( f )] = N P ( f )
2
()
From Equation () the bandwidth required from a sequence of pulses is basically
the same as the bandwidth of a single pulse. From Figure (3.11) the main lobe
width is 2rb.
Sec. 3.3 Fourier Transform
21
3.3.3 Fourier Transform of a Sampled Signal
v
A sampled signal x(m) can be modelled as
x ( m) =
∞
∑ x(t )δ (t − mT )
s
m = −∞
(3.54)
where m is the discrete time variable and Ts is the sampling period. The Fourier
transform of x(m), a sampled version of a continuous signal x(t), can be obtained
from Eq. (3.21) as
∞
X ( f )=
∞
∑ x(t )δ (t − m)e − j 2πft dt =
∫
−∞ m = −∞
=
∞
∑ x(mT )e
m = −∞
s
∞
∞
∑ ∫ x(t )δ (t − mT )e
m = −∞ −∞
s
− j 2πmf / Fs
− j 2πft
dt
(3.55)
For convenience of notation and without loss of generality it is often assumed that
sampling frequency Fs=1/Ts=1, hence
Xs( f ) =
∞
∑ x(m)e
− j 2πmf
(3.56)
m = −∞
The inverse Fourier transform of a sampled signal is defined as
1/ 2
x ( m) =
∫X
s
( f ) e j2πfm df
(3.57)
−1 / 2
Note that x(m) and X s ( f ) are equivalent in that they contain the same
information in different domains. In particular, as expressed by the Parseval's
theorem, the energy of the signal may be computed either in the time or in the
frequency domain a
Signal Energy =
∞
1/ 2
m = −∞
−1 / 2
∑ x 2 (m) = ∫ | X s ( f ) |2 df
(3.58)
Example 3.10 Show that the spectrum of a sampled signal is periodic with a
period equal to the sampling frequency Fs.
22
Chap.3 Fourier Analysis and Synthesis
Solution: substitute f+kFs for the frequency variable f in Eq. (3.56)
X ( f + kFs ) =
∞
∑ x(mT )e
s
− j2πm
( f + kFs )
Fs
∞
=
∑ x(mT )e
− j2πm
s
m = −∞
m = −∞
f
Fs
− j2πm
kFs
Fs
e1
424
3= X(f )
=1
(3.59)
Fig. 3.13.a shows the spectrum of a band-limited continuous-time signal. As
shown in Fig. 3.13.b after the signal is sampled its spectrum becomes periodic.
Xs(f)
X(f)
...
...
0
-2Fs
f
-Fs
(a)
0
Fs
2Fs
f
(b)
Figure 3.13 The spectrum of : (a) a continuous signal, and (b) its sampled version.
x(0)
x(1)
x(2)
x(N–2)
x(N – 1)
Discrete Fourier
Transform
.
.
.
N–1
X(k) =
∑
m=0
2 π kn
x(m) e N
–j
.
.
.
X(0)
X(1)
X(2)
X(N – 2)
X(N– 1)
Figure 3.14 Illustration of the DFT as a parallel-input parallel-output signal processor.
Sec. 3.5 Short-Time Fourier Transform
23
3.4 Discrete Fourier Transform (DFT)
When a non-periodic signal is sampled, its Fourier transform becomes a periodic
but continuous function of frequency, as shown in Eq. (3.59). The discrete Fourier
transform (DFT) is derived from sampling the Fourier transform of a discrete-time
signal. For a finite duration discrete-time signal x(m) of length N samples, the
discrete Fourier transform (DFT) is defined as N uniformly spaced spectral
samples
N −1
X ( k ) = ∑ x ( m) e
−j
2π
mk
N
k = 0, . . ., N−1
(3.60)
m=0
Comparing Eqs. (3.60) and (3.56) we see that the DFT consists of N equi-spaced
samples taken from one period (2π) of the continuous spectrum of the discrete
time signal x(m). The inverse discrete Fourier transform (IDFT) is given by
2π
x ( m) =
j mk
1 N −1
X (k ) e N
∑
N k =0
m= 0, . . ., N−1
(3.61)
A periodic signal has a discrete spectrum. Conversely any discrete frequency
spectrum belongs to a periodic signal. Hence the implicit assumption in the DFT
theory, is that the input signal x(m) is periodic with a period equal to the
observation window length of N samples.
Example: Derivation of inverse discrete Fourier transform Obtain the inverse
DFT from the DFT equation. The discrete Fourier transform (DFT) is given
by
N −1
X ( k ) = ∑ x ( m) e
−j
2π
km
N
k = 0, . . ., N−1
m =0
Multiply both sides of the DFT equation by e-j2πkn/N and take the summation
as
24
Chap.3 Fourier Analysis and Synthesis
N −1
∑ X (k ) e
+j
2π
kn
N
k =0
=
N −1 N −1
∑∑ x(m) e
−j
2π
km
N
e
+j
2π
kn
N
k =0 m=0
N −1
N −1
−j
2π
k ( m−n)
= ∑ x ( m) ∑ e N
k =0
m=0
14
4244
3
N if m = n
0 otherwise
Using the orthogonality principle,the inverse DFT equation can be derived
as
2π
−j
km
1 N −1
x ( m) = ∑ X ( k ) e N
N k =0
3.4.1 Time and Frequency Resolutions: The Uncertainty Principle
Signals such as speech, music or image are composed of nonstationary  i.e.
time-varying and/or space varying  events. For example speech is composed of
a string of short-duration sounds called phonemes, and an image is composed of
various objects. When using the DFT it is desirable to have a high enough time
and space resolution in order to obtain the spectral characteristics of each
individual elementary event or object in the input signal. However there is a
fundamental trade-off between the length, i.e. the time or space resolution, of the
input signal and the frequency resolution of the output spectrum. The DFT takes
as the input a window of N uniformly spaced time domain samples [x(0), x(1), …,
x(N−1)] of duration ∆T=N.Ts, and outputs N spectral samples [X(0), X(1), …,
X(N−1)] spaced uniformly between zero Hz and the sampling frequency Fs=1/Ts
Hz. Hence the frequency resolution of the DFT spectrum ∆f, i.e. the space between
successive frequency samples, is given by
∆f =
F
1
1
=
= s
∆T NTs N
(3.62)
Note that the frequency resolution ∆f and the time resolution ∆T are inversely
proportional in that they can not both be simultanously increased, in fact ∆T∆f=1.
This is known as the uncertainty principle.
Example 3.11 A DFT is used in a DSP system for the analysis of an analog
signal with a frequency content of up to 10 kHz. Calculate: (i) the minimum
sampling rate Fs required, and (ii) the number of samples required for the DFT to
achieve a frequency resolution of 10 Hz at the minimum sampling rate.
Sec. 3.5 Short-Time Fourier Transform
25
Solution:
(i)
Sampling rate > 2×10 kHz, say 22 kHz, and
∆f =
(ii)
Fs
N
10 =
22000
N
N ≥ 2200 .
Example 3.12 Write a MATLAB program to explore the spectral resolution of a
signal consisting of two sinewaves, closely spaced in frequency, with the varying
length of the observation window.
Solution:
In the following program the two sinwaves have frequencies of 100 Hz and 110
Hz, and the sampling rate is 1 kHz. We experiment with two time windows of
length N1=1024 with a theoretical frequency resolution of ∆f=1000/1024=0.98 Hz,
and N2=64 with a theoretical frequency resolution ∆f=1000/64=15.7 Hz.
Amplitude
2
1
0
- 1
- 2
2 0 0
0
6 0 0
4 0 0
1 0 0 0
8 0 0
T im e
(a)
450
Amplitude
Amplitude
80
60
300
40
150
20
0
10
20
30
40
50
60
70
0
100
200
300
(b)
400
500
600
Feq k
Feq k
(c)
Figure 3.15 Illustration of time and frequency resolutions: (a) sum of two sinewaves with 10
Hz difference in their frequencies, (b) the spectrum of a segment of 64 samples from
demonstrating insufficient frequency resolution to separate the sinewaves, (c) the spectrum of
a segment of 1024 samples has sufficient resolution to show the two sinewaves.
Fs=1000; F1=100; F2=110;
N=1:1024; N1=1024; N2=64;
x1=sin(2*pi*F1*N/Fs); x2=sin(2*pi*F2*N/Fs); y=x1+x2;
26
Chap.3 Fourier Analysis and Synthesis
Y1=abs(fft(y(1:N1)));Y2=abs(fft(y(1:N2)));
figure(1); plot(y); figure(2); plot(Y1(1:N1/2)); figure(3); plot(Y2(1:N2/2));
3.4.2 The Effect of Finite Length Data on DFT (Windowing)
In a practical situation we have either with a short length signal, or with a long
signal of which the DFT can only handle one segment at a time. Having a short
segment of N samples of a signal or taking a slice of N samples from a signal is
equivalent to multiplying the signal by a unit-amplitude rectangular pulse window
of N samples. Therefore an N-sample segment of a signal x(m) is equivalent to
x w (m) = w(m) x(m)
(3.63)
where w(m) is a rectangular pulse of N samples duration given is
1 0≤m ≤ N − 1
w(m)=
0 otherwise
(3.64)
Multiplying two signals in time is equivalent to the convolution of their frequency
spectra. Thus the spectrum of a short segment of a signal is convolved with the
spectrum of a rectangular pulse as
X w (k ) = W (k ) * X (k )
(3.65)
The result of this convolution is some spreading of the signal energy in the
frequency domain as illustrated in the next example.
Example 3.13 Find the DFT of a rectangular window given by
1 0 ≤ m ≤ N − 1
w(m)=
0 otherwise
(3.66)
Solution: Taking the DFT of w(m), and using the convergence formula for the
partial sum of a geomertic series, described in the appendix A, we have
W (k ) =
N −1
∑
m =0
w(m)e
−j
2π
mk
N
=
1 − e − j2πk
1− e
−j
2π
k
N
=e
−j
( N −1)
πk
N
sin(πk )
sin(πk / N )
(3.67)
Note that for the integer values of k, w(k) is zero except for k=0.
Example 3.14 Find the spectrum of an N-sample segment of a complex sinewave
with a fundamental frequency F0=1/T0.
Sec. 3.5 Short-Time Fourier Transform
27
Solution: Taking the DFT of x(m) = e − j 2πF0 m we have
X (k ) =
N −1
∑e
− j2πF0 m
m =0
e
−j
2π
mk
N
=
N −1 − j2π ( F − k ) m
0
N
e
∑
m =0
−j
1 − e − j2π ( NF0 − k )
=
=e
− j2π ( NF0 − k ) / N
1− e
( N −1)
π ( NF0 − k )
N
sin(π ( NF0 − k ))
sin(π ( NF0 − k ) / N )
(3.68)
Note that for integer values of k, X(k) is zero at all samples but one k=0.
3.4.3 End-Point Effects in DFT; Spectral Energy Leakage and
Windowing
In DFT the input signal is assumed to be periodic, with a period equal to the length
of the observation window of N samples. Now, for a sinusoidal input signal if
there is an integer number of cycles within the observation window, as in Fig.
3.16.a, then the assumed periodic waveform is the same as an infinite length pure
sinusoid. But if the observation window contains a non-integer number of cycles
of a sinusoid then the assumed periodic waveform will not be a pure sine wave and
will have end-point discontinuities. The spectrum of the signal then differs from
the spectrum of sinewave as illustrated in Fig. 3.16.b. The overall effects of finite
length window and end-point discontinuities are:
time
time
2 cycles
F0
(a)
2.25 cycles
Frequency
F0
Frequency
(b)
Figure 3.16 The DFT spectrum of exp(j2πfm) : (a) an integer number of cycles with in the
N-sample analysis window, (b) a non-integer number of cycles in the window.
1. The spectral energy which could have been concentrated at a single point, or
in a narrow band of frequencies is spread over a larger band of frequencies.
28
Chap.3 Fourier Analysis and Synthesis
2. A smaller amplitude signal, located in frequency near a larger amplitude
signal, may be obscured by one of the larger signal’s side-lobes. That is sidelobes of a large amplitude signal may interfere with the main-lobe of a nearby
small amplitude signal.
The end-point problems may be alleviated using a window that gently drops to
zero. One such window is a raised cosine window of the form
2πm

0≤m≤ N − 1
α − (1 − α ) cos
w(m) = 
N

0
otherwise
(3.69)
For α=0.5 we have the Hanning window also known as the raised cosine window
2πm
N
For α=0.54 we have the Hamming window
wHan (m) = 0.5−0.5cos
wHam (m) = 0.54 − 0.46 cos
2πm
N
0≤m≤ N − 1
(3.70)
0 ≤ m ≤ N −1
(3.71)
0
-5
-1 0
-1 5
-2 0
-2 5
0
T im e
6 4
-3 0
0
F re q u en c y
F
s
/2
0
-1 0
-2 0
-3 0
-4 0
-5 0
-6 0
0
0
6 4
F
s
/2
0
-1 0
-2 0
-3 0
-4 0
-5 0
0
6 4
-6 0
-7 0
-8 0
0
F
s
/2
Figure 3.17 (a) Rectangular window frequency response, (b) Hamming window frequency
response, (c) Hanning window frequency response.
3.4.3 Spectral Smoothing
Sec. 3.5 Short-Time Fourier Transform
29
The spectrum of a short length signal can be interpolated to obtain a smoother
spectrum. Interpolation of the frequency spectrum X(k) is achieved by zeropadding of the time domain signal x(m). Consider a signal of length N samples
[x(0), . . ., x(N−1)]. Increase the signal length from N to 2N samples by padding N
zeros to obtain the padded sequence [x(0), . . ., x(N−1), 0, . . ., 0]. The DFT of the
padded signal is given by
2 N −1
X (k ) =
∑ x ( m) e
−j
N −1
−j
2π
mk
2N
m=0
=
∑ x ( m) e
π
N
k = 0, . . ., 2N−1
(3.72)
mk
m =0
The spectrum of the zero-padded signal, Eq. (3.72), is composed of 2N spectral
samples; N of which, [X(0), X(2), X(4), X(6), . . . X(2N−2)] are the same as those
that would be obtained from a DFT of the original N samples, and the other N
samples [X(1), X(3), X(5), X(6), . . . X(2N−1)] are interpolated spectral lines that
result from zero-padding. Note that zero padding does not increase the spectral
resolution, it merely has an interpolating or smoothing effect in the frequency
domain, as illustrated in Fig 3.18.
r(m)
|R(f)|
without zero padding
Padded zero’s
Time
|R(f)|
Frequency
|R(f)|
with 20 padded zeros
with 10 padded zeros
Frequency
Frequency
Figure 3.18 Illustration of the interpolating effect, in the frequency domain, of zero
padding a signal in the time domain.
30
Chap.3 Fourier Analysis and Synthesis
time
Figure 3.19 - Segmentation of speech using Hamming window for STFT.
3.5 Short-Time Fourier Transform
In Fourier transform it is assumed that the signal is stationary, meaning the signal
statistics, such as the mean, the power, and the power spectrum, are time-invariant.
Most real life signals such as speech, music and image signals are nonstationary in
that their amplitude, power, spectral composition and other features changes
continuously with time. To apply Fourier transform to nonstationary signals the
signal is divided into appropriately short-time windows, such that within each
window the signal can be assumed to be time-invariant. The Fourier transform
applied to the short signal segment within each window is known as the short-time
Fourier Transform. Fig. 3.19 illustrates the segmentation of a speech signal into a
sequence of overlapping, hamming windowed, short segments. The choice of
window length is a compromise between the time resolution and the frequency
resolution. For Audio signals a time window of about 25 ms, corresponding to a
frequency resolution of 40 Hz is normally adopted.
walker.qxp 3/24/98 2:16 PM Page 658
Fourier Analysis
and Wavelet
Analysis
James S. Walker
I
n this article we will compare the classical
methods of Fourier analysis with the newer
methods of wavelet analysis. Given a signal, say a sound or an image, Fourier analysis easily calculates the frequencies and
the amplitudes of those frequencies which make
up the signal. This provides a broad overview of
the characteristics of the signal, which is important for theoretical considerations. However,
although Fourier inversion is possible under
certain circumstances, Fourier methods are not
always a good tool to recapture the signal, particularly if it is highly nonsmooth: too much
Fourier information is needed to reconstruct
the signal locally. In these cases, wavelet analysis is often very effective because it provides a
simple approach for dealing with local aspects
of a signal. Wavelet analysis also provides us with
new methods for removing noise from signals
that complement the classical methods of
Fourier analysis. These two methodologies are
major elements in a powerful set of tools for theoretical and applied analysis.
This article contains many graphs of discrete
signals. These graphs were created by the computer program FAWAV, A Fourier–Wavelet Analyzer, being developed by the author.
James S. Walker is professor of mathematics at the
University of Wisconsin-Eau Claire. His e-mail address
is walkerjs@uwec.edu.
The author would like to thank Hugo Rossi, Steven
Krantz, his colleague Marc Goulet, and two anonymous reviewers for their helpful comments during the
writing of this article.
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Frequency Information, Denoising
As an example of the importance of frequency information, we will examine how Fourier analysis can
be used for removing noise from signals. Consider
a signal f (x) defined over the unit interval (where here
x stands for time). The periodP1 Fourier series exi2π nx , with
pansion
n∈Z cn e
R 1 of f is defined by
cn = 0 f (x)e−i2π nx dx. Each Fourier coefficient, cn , is
an amplitude associated with the frequency n of the
exponential ei2π nx. Although each of these exponentials has a precise frequency, they all suffer from
a complete absence of time localization in that their
magnitudes, |ei2π nx | , equal 1 for all time x .
To see the importance of frequency information,
let us examine a problem in noise removal. In
Figure 1(a)[top] we show the graph of the signal
(1)
f (x) =
2
(5 cos 2π νx) [ e−640π (x−1/8)
2
2
+ e−640π (x−3/8) + e−640π (x−4/8)
2
2
+ e−640π (x−6/8) + e−640π (x−7/8) ]
where the frequency, ν , of the cosine factor is 280 .
Such a signal might be used by a modem for transmitting the bit sequence 1 0 1 1 0 1 1 . The Fourier coefficients for this signal are shown in Figure 1(b)[top].
The highest magnitude coefficients are concentrated
around the frequencies ±280. Suppose that when this
signal is received, it is severely distorted by added
noise; see Figure 1(a)[middle]. Using Fourier analysis, we can remove most of this noise. Computing the
noisy signal’s Fourier coefficients, we obtain the
graph shown in Figure 1(b)[middle]. The original signal’s largest magnitude Fourier coefficients are clusAMS
VOLUME 44, NUMBER 6
walker.qxp 3/24/98 2:16 PM Page 659
Figure 1. (a)[top] Signal. (b)[top] Fourier coefficients of signal. (a)[middle] Signal after adding noise.
(b)[middle] Fourier coefficients of noisy signal and filter function. (b)[bottom] Fourier coefficients
after multiplication by filter function. (a)[bottom] Recovered signal.
tered around the frequency positions ±280. The
Fourier coefficients of the added noise are localized around the origin, and they decrease in
magnitude until they are essentially zero near
the frequencies ±280 . Thus, the original signal’s coefficients and the noise’s coefficients
are well separated. To remove the noise from the
signal, we multiply the noisy signal’s coefficients
by a filter function, which is 1 where the signal’s
coefficients are concentrated and 0 where the
noise’s coefficients are concentrated. We then recover essentially all of the signal’s coefficients;
see Figure 1(b)[bottom]. Performing a Fourier
series partial sum with these recovered coefficients, we obtain the denoised signal, which is
shown in Figure 1(a)[bottom]. Clearly, the bit sequence 1 0 1 1 0 1 1 can now be determined from
the denoised signal, and the denoised signal is
a close match of the original signal. In the section “Signal Denoising” we shall look at another
example of this method and also discuss how
wavelets can be used for noise removal.
Signal Compression
As the example above shows, Fourier analysis is
very effective in problems dealing with frequency
location. However, it is often very ineffective at
representing functions. In particular, there are
severe problems with trying to analyze transient signals using classical Fourier methods.
For example, in Figure 2(a)[top] we show a discrete signal obtained from M = 1024 values
{fj = F(j/M)}M−1
of the function F(x) =
j=0
5
for n = − 12 M + 1, . . . , 0, . . . , 12 M . The discrete
Fourier coefficients f̂n can be calculated by a fast
Fourier transform (FFT) algorithm and are the
discrete analog of the Fourier coefficients cn
for F , when fj = F(j/M) . Moreover, f̂n is just a
Riemann sum approximation of the integral that
defines cn .1 The magnitudes of the discrete
Fourier coefficients for this transient damp down
to zero very slowly (their graph is a very wide
bell-shaped curve with maximum at the origin).
Consequently, to represent the transient well, one
must retain most if not all of these Fourier coefficients. In Figure 2(a)[bottom] we show the results obtained from trying to compress the transient
a discrete partial sum
P104 by computing
i2π nj using only one-fifth of the
n=−104 f̂n e
Fourier coefficients.2 Clearly, even a moderate
compression ratio of 5:1 is not effective.
Wavelets, however, are often very effective at
representing transients. This is because they are
designed to capture information over a large
range of scales. A wavelet series expansion of a
function f is defined by
X
n/2
βn
ψ (2n x − k)
k2
n,k∈Z
with
βn
k =
Z∞
−∞
f (x) 2n/2 ψ (2n x − k) dx .
The function ψ (x) is called the wavelet, and the
coefficients βn
k are called the wavelet coefficients. The function 2n/2 ψ (2n x − k) is the
2
e−10 π (x−.6) . For this example, we compute the
discrete Fourier series coefficients f̂n defined by
f̂n = M
JUNE/JULY 1997
P
−1 M−1
j=0
fj
e−i2π nj/M
1For further discussion of discrete Fourier coefficients,
see [2] or [11].
P512
i2π nj , which uses all of the disn=−511 f̂n e
crete Fourier coefficients, equals fj.
2The sum
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Figure 2. (a)[top] Signal. (a)[bottom] Fourier series using 205 coefficients, 5:1 compression. (b)[top]
Wavelet coefficients for signal. (b)[bottom] Wavelet series using only largest 4% in magnitude of
wavelet coefficients, 25:1 compression.
wavelet shrunk by a factor of 2n if n is positive
(magnified by a factor of 2−n if n is negative) and
shifted by k2−n units. The factor 2n/2 in the expression 2n/2 ψ (2n x − k) preserves the L2-norm.
Since the wavelet series depends on two parameters of scale and translation, it can often be
very effective in analyzing signals. These parameters make it possible to analyze a signal’s
behavior at a dense set of time locations and with
respect to a vast range of scales, thus providing
the ability to zoom in on the transient behavior
of the signal. For example, let us examine the earlier transient using a discretized version of a
wavelet series. We shall use a Daubechies order
4 wavelet (Daub4 for short; see the section
“Daubechies Wavelets”). In Figure 2(b)[top] we
show all of the 1024 wavelet coefficients of this
transient and observe that most of these coefficients are close to 0 in magnitude. Consequently, by retaining only the largest magnitude
coefficients for use in a wavelet series, we obtain significant compression. In Figure 2(b)[bottom] we show the reconstruction of the transient
using only the top 4% in magnitude of the wavelet
coefficients, a 25:1 compression ratio. Notice
how accurately the transient is represented. In
fact, the maximum error at all computed points
is less than 9.95 × 10−14 . There is an important
application here to the field of signal transmission. By transmitting only these 4% of the wavelet
coefficients, the information in the signal can be
transmitted 25 times faster than if we transmitted all of the original signal. This provides a considerable boost in efficiency of transmission.
We shall look at more examples of compression in the section “Compression of Signals”, but first we shall describe how wavelet
analysis works.
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The Haar Wavelet
In order to understand how wavelet analysis
works, it is best to begin with the simplest
wavelet, the Haar wavelet. Let 1A (x) denote the
indicator function of the set A , defined by
1A (x) = 1 if x ∈ A and 1A (x) = 0 if x ∈
/ A . The
ψ is defined by
Haar
wavelet
ψ (x) = 1[0, 1 ) (x) − 1[ 1 ,1) (x) . It is 0 outside of
2
2
[0, 1), so it is well localized in time, and it satisfies
Z∞
Z∞
ψ (x) dx = 0,
|ψ (x)|2 dx = 1.
−∞
−∞
The Haar wavelet ψ (x) is closely related to the
function φ(x) defined by φ(x) = 1[0,1) (x) . This
function φ(x) is called the Haar scaling function.
Clearly, the Haar wavelet and scaling function
satisfy the identities
ψ (x) = φ(2x) − φ(2x − 1),
(2)
φ(x) = φ(2x) + φ(2x − 1),
and the scaling function satisfies
Z∞
−∞
φ(x) dx = 1,
Z∞
−∞
|φ(x)|2 dx = 1.
The Haar wavelet ψ (x) generates the system
of functions {2n/2 ψ (2n x − k)} . It is possible to
show directly that {2n/2 ψ (2n x − k)} is an orthonormal basis for L2 (R), but it is more illuminating to put the discussion on an axiomatic
level. This axiomatic approach leads to the
Daubechies wavelets and many other wavelets
as well. We begin by defining the subspaces
{Vn }n∈Z of L2 (R) in the following way:
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walker.qxp 3/24/98 2:17 PM Page 661
n
Vn = step functions in L2 (R), constant
o
k k+1
,
),
k
∈
Z
.
2n 2n
This set of subspaces {Vn }n∈Z satisfies the following five axioms [6]:
on the intervals [
Axioms for a Multi-Resolution Analysis
(MRA)
Scaling: f (x) ∈ Vn if and only if f (2x) ∈ Vn+1.
Inclusion: Vn ⊂ Vn+1, for each n.
S
Density: closure
Vn = L2 (R) .
n∈Z
T
Vn = {0}.
Maximality:
n∈Z
Basis: ∃φ(x) such that {φ(x − k)}k∈Z is an
orthonormal basis for V0 .
To satisfy the basis axiom, we shall use the
Haar scaling function φ defined above. Then, by
combining the scaling axiom with the basis
axiom, we find that {2n/2 φ(2n x − k)}k∈Z is an
orthonormal basis for Vn . But the totality of all
these orthonormal bases, consisting of the set
{2n/2 φ(2n x − k)}k,n∈Z, is not an orthonormal
basis for L2 (R) because the spaces Vn are not
mutually orthogonal. To remedy this difficulty,
we need what are called wavelet subspaces. Define the wavelet subspace Wn to be the orthogonal complement of Vn in Vn+1 . That is, Wn satisfies the equation Vn+1 = Vn ⊕ Wn where ⊕
denotes the sum of mutually orthogonal subspaces. From the density axiom and repeated application ofLthe last equation, we obtain
L2 (R) = V0 ⊕ ∞
V0 in a
n=0 Wn . Decomposing
L
similar way, we obtain L2 (R) = n∈Z Wn . Thus,
L2 (R) is an orthogonal sum of the wavelet subspaces Wn.
Using (2) and the MRA axioms, it is easy to
prove the following lemma.
Note that these wavelets have period 1 . Furthermore,
ψ̃n,k ≡ 0 for n < 0 , and
ψ̃n,k+2n = ψ̃n,k for all k ∈ Z and n ≥ 0. On the interval [0, 1), the periodic Haar wavelets ψ̃n,k satisfy ψ̃n,k (x) = 2n/2 ψ (2n x − k) for n ≥ 0 and
k = 0, 1, . . . , 2n − 1. So we have the following theorem as a consequence of Theorem 1.
Theorem 2. The functions 1 and ψ̃n,k for n ≥ 0
and k = 0, 1, . . . , 2n − 1 are an orthonormal basis
for L2 [0, 1) .
Remark. In the section “Daubechies Wavelets”
we will make use of periodized scaling functions, φ̃n,k, defined by
(4)
Fast Haar Transform
The relation between the Haar scaling function
φ and wavelet ψ leads to a beautiful set of relations between their coefficients as bases. Let
{αkn } and {βn
k } be defined by
Z∞
αkn =
f (x) 2n/2 φ(2n x − k) dx,
−∞
Z∞
(5)
βn
=
f (x) 2n/2 ψ (2n x − k) dx.
k
−∞
Substituting 2n x in place of x in the identities
in (2), we obtain
n+1
1
2n/2 φ(2n x) = √ [2 2 φ(2n+1 x)]
2
n+1
1
+ √ [2 2 φ(2n+1 x − 1)]
2
n+1
1
2n/2 ψ (2n x) = √ [2 2 φ(2n+1 x)]
2
n+1
1
− √ [2 2 φ(2n+1 x − 1)].
2
It then follows that
{2n/2 ψ (2n x − k)}k,n∈Z
(3)
ψ̃n,k (x) =
X
j∈Z
JUNE/JULY 1997
2n/2 ψ 2n (x + j) − k .
2n/2 φ 2n (x + j) − k
for n ≥ 0 and k = 0, 1, . . . , 2n − 1.
It follows from the scaling axiom that
{2n/2 ψ (2n x − k)}k∈Z is an orthonormal basis
for Wn. Therefore, since L2 (R) is the orthogonal
sum of all the wavelet subspaces Wn, we have
obtained the following result.
are an orthonormal basis for L2 (R).
This orthonormal basis is the Haar basis for
L2 (R). There is also a Haar basis for L2 [0, 1). To
obtain it, we first define periodic wavelets ψ̃n,k
by
X
j∈Z
Lemma 1. The functions {ψ (x − k)}k∈Z are an
orthonormal basis for the subspace W0 .
Theorem 1. The functions
φ̃n,k (x) =
(6)
1 n+1
+
αkn = √ α2k
2
1 n+1
√ α2k
βn
−
k =
2
1 n+1
√ α2k+1
,
2
1 n+1
√ α2k+1
.
2
This result shows that the nth level coefficients
st
αkn and βn
k are obtained from the (n + 1) level
n+1
coefficients αk through multiplication by the
following orthogonal matrix:
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AMS
661
walker.qxp 3/24/98 2:17 PM Page 662


O=
(7)
√1
2
√1
2
√1
2
−1
√
2

cient α00 and a single wavelet coefficient β00 ,
and at this last step the permutation P2 is unnecessary. The complete transformation, denoted by H , satisfies

.
Successively applying this orthogonal matrix O ,
we obtain {αkn } and {βn
k } starting from some
highest level coefficients {αkN } for some large
N . Because of the density axiom, N can be chosen
enough to approximate f by
P large
N N/2
φ(2N x − k) in L2-norm as closely
k∈Z αk 2
as desired.
Let us now discretize these results. Suppose
that we are working with data {fj }M−1
j=0 associM−1
ated with the time values {xj = j/M}j=0
on the
unit interval. If we shrink the Haar scaling function φ(x) = 1[0,1) (x) enough, it covers only the
first point, x0 = 0 . Consequently, by choosing a
large enough N , we may assume that our scaling coefficients {αkN } satisfy αkN = fk , for
k = 0, 1, . . . , M − 1 . Assuming that M is a power
of 2 , say M = 2R , it follows that N = R .
The next step involves expressing the coefficient relations in (6) in a matrix form. Let A ⊕ B
stand for the orthogonal
sum
of the matrices A
A0
and B, that is, A ⊕ B = 0 B . Now let HM denote
the M × M orthogonal matrix defined by
HM = O ⊕ O ⊕ · · · ⊕ O , where the orthogonal
matrix sums are applied M/2 times and O is the
matrix defined in (7). Then, by applying the coefficient relations in (6) and using the fact that
fk = αkR, we obtain
HM [f0 , f1 , . . . , fM−1 ]T
h
= α0R−1 , βR−1
, α1R−1 , βR−1
,
0
1
T
, βR−1
.
. . . , αR−1
1
1
2
M−1
2
2
M−1
...
, αR−1
}
1
M−1
2
2
Discrete Haar Series
The fast Haar transform can be used for computing partial sums of the discretized version
of the following Haar wavelet series in L2 [0, 1) :
M−1
4
} . These
scaling coefficients {α0R−2 , . . . , αR−2
1
M−1
operations continue until we can no longer
divide the number of components by 2 . At the
R = log2 M step, we obtain a single scale coeffiNOTICES
α00 +
∞ 2X
−1
X
n/2
βn
ψ (2n x − k).
k2
Let us assume, as in the previous section, that
we have a discrete signal {fj }M−1
j=0 associated
with the time values {xj = j/M}M−1
j=0 on the unit
interval. Substituting these time values into (8)
and restricting the upper limit of n, we obtain
OF THE
α00 CM
+
n
R−1
−1
X 2X
n/2
βn
ψ (2n xj − k).
k CM 2
n=0 k=0
R−2
R−2
wavelet coefficients {β0 , . . . , β 1 M−1 } and
662
Therefore, the inverse operation is also an O( M )
operation.
fj =
to obtain the next level
4
· · · (H2T ⊕ IM−2 ).
M−1
If we go to the next lower level, the transformations just described are repeated, only now
the matrices used are HM/2 and PM/2 , and they
operate only on the M/2 -length vector
{α0R−1 ,
T T
T
T
H −1 = HM
PM (HM/2
⊕ IM/2 ) (PM/2
⊕ IM/2 )
n=0 k=0
T
R−1
R−1
= α0R−1 , . . . , αR−1
,
β
,
.
.
.
,
β
.
1
1
0
2
where IN is the N × N identity matrix.
These matrix multiplications can be performed rapidly on a computer. Multiplying by
HM requires only O(M) operations, since HM
consists mostly of zeroes. Similarly, the permutation PM requires O(M) operations. Therefore, the whole transformation requires
O(M) + O(M/2) + · · · + O(2) = O(M) operations.
The transformation H is called a fast Haar
transform. It should be noted that FFTs, which
have revolutionized scientific practice during
the last thirty years, are O(M log M) algorithms.
Since each Hk is an orthogonal matrix, and
so is every permutation Pk , it follows that H is
invertible. Its inverse is
n
To sort the coefficients properly into two groups,
we apply an M × M permutation matrix PM as
follows:
T
R−1
R−1
PM α0R−1 , βR−1
,
.
.
.
,
α
,
β
1
1
0
M−1
· · · (PM/2 ⊕ IM/2 ) (HM/2 ⊕ IM/2 ) PM HM
(8)
M−1
2
H = (H2 ⊕ IM−2 )
AMS
The right side of this equation is just the transformation H −1 H applied to {fj } . The first
part of this transformation, H {fj }, produces the
0
0
1
1
R−1
, and the
coefficients α0 , β0 , β0 , β1 , . . . , β 1
2
M−1
second part, the application of H −1 , reproduces the original data {fj } . The constant CM
is a scale factor which ensures that the constant
vector CM and the vectors {CM 2n/2 ψ (2n xj − k)}
are unit vectors in RM , using the
p standard inner
product. Consequently, CM = 1/M .
There are many ways of forming partial sums
of discretized Haar series. The simplest ones conVOLUME 44, NUMBER 6
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Figure 3. (a)[top] Haar series partial sum, 229 terms. (b)[top] Fourier series partial sum, 229 terms.
(a)[bottom] Haar series partial sum, 92 terms. (b)[bottom] Daub4 series partial sum, 22 terms.
sist of multiplying the data by H , then setting
some of the resulting coefficients equal to 0 , and
then multiplying by H −1 . A widely used method
involves specifying a threshold. All coefficients
whose magnitudes lie below this threshold are
set equal to 0 . This method is frequently used
for noise removal, where coefficients whose
magnitudes are significant only because of the
added noise will often lie below a well-chosen
threshold. We shall give an example of this in the
section “Signal Denoising”. A second method
keeps only the largest magnitude coefficients,
while setting the rest equal to 0 . This method
is convenient for making comparisons when it
is known in advance how many terms are needed.
We used it in the compression example in the section “Signal Compression”. A third method,
which we shall call the energy method, involves
specifying a fraction of the signal’s energy, where
the energy is the square root of the sum of the
squares of the coefficients.3 We then retain the
least number of the largest magnitude coefficients whose energy exceeds this fraction of the
signal’s energy and set all other coefficients
equal to 0 . The energy method is useful for theoretical purposes: it is clearly helpful to be able
to specify in advance what fraction of the signal’s energy is contained in a partial sum. We
shall use the energy method frequently below.
Let us look at an example. Suppose our signal is
{fj = F(j/8192)}8191
j=0
where F(x) = x1[0,.5) (x) + (x − 1)1[.5,1) (x) . In Figure 3(a)[top] we show a Haar series partial sum,
3The Haar transform is orthogonal, so it makes sense
to specify energy in this way.
JUNE/JULY 1997
created by the energy method, which contains
99.5% of the energy of this signal. This partial
sum, which used 229 coefficients out of a possible 8192, provides an acceptable visual representation of the signal. In fact, the sum of the
squares of the errors is 2.0 × 10−3 . By comparison a 229 coefficient Fourier series partial sum
suffers from serious drawbacks (see Figure
3(b)[top]). The sum of the squares of the errors
is 4.5 × 10−1 , and there is severe oscillation and
a Gibbs’ effect near x = 0.5 . Although these latter two defects could be ameliorated using other
summation methods ([11], Ch. 4), there would
still be a significant deviation from the original
signal (especially near x = 0.5 ).
This example illustrates how wavelet analysis homes right in on regions of high variability
of signals and that Fourier methods try to
smooth them out. The size of a function’s Fourier
coefficients is related to the frequency content
of the function, which is measured by integration of the function against completely unlocalized basis functions. For a function having a
discontinuity, or some type of transient behavior, this produces Fourier coefficients that decrease in magnitude at a very slow rate. Consequently, a large number of Fourier coefficients
are needed to accurately represent such signals.4 Wavelet series, however, use compactly
supported basis functions which, at increasing
levels of resolution, have rapidly decreasing supports and can zoom in on transient behavior. The
transient behaviors contribute to the magnitude
of only a small portion of the wavelet coefficients.
4In recent years, though, significant improvements
have been achieved using local cosine bases [3, 7,
9, 5, 1].
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Figure 4. (a) Magnitudes of highest level coefficients for a function F: [top] Haar coefficients, [middle]
Daub4 coefficients; [bottom] Graph of F. (b) Sums of squares of all coefficients: [top] Haar coefficients,
[middle] Daub4 coefficients; [bottom] Graph of F. Vertical scales for highest level coefficients and sums
of squares are logarithmic.
Consequently, a small number of wavelet coefficients are needed to accurately represent such signals.
The Haar system performs well when the signal
is constant over long stretches. This is because the
Haar wavelet is supported on
R ∞[0, 1) and satisfies a
0th order moment condition, −∞ ψ (x) dx = 0 . Therefore, if the signal {fj } is constant over an interval
a ≤ xj < b such that [k2−n , (k + 1)2−n ) ⊂ [a, b) , then
n
the wavelet coefficient βk equals 0 . For example, suppose {fj = F(j/8192)}8191
j=0 where
F(x) = (8x − 1)1[.125,.25) (x)
+ 1[.25,.75) (x) + (7 − 8x)1[.75,.875) (x).
In Figure 3(a)[bottom] we show a Haar series partial
sum which contains 99.5% of the energy of this signal and uses only 92 coefficients out of a possible
8192. The fact that the signal is constant over three
large subintervals of [0, 1) accounts for the excellent
compression in this example. In order to obtain
wavelet bases that provide considerably more compression, we need a compactly supported wavelet
ψ (x) which has more moments equal to zero. That
is, we want
(9)
Z∞
−∞
xj ψ (x) dx = 0, for j = 0, 1, . . . , L − 1
for an integer L ≥ 2 . We say that such a wavelet has
its first L moments equal to zero. For example, a
Daub4 wavelet has its first 2 moments equal to zero.
Using a Daub4 wavelet series for the signal above, it
is possible to capture 99.5% of the energy using only
22 coefficients! See Figure 3(b)[bottom]. This improvement in compression is due to the fact that the
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Daub4
R ∞wavelet is supported
R ∞ on [0, 3) and satisfies −∞ ψ (x) dx = 0 and −∞ xψ (x) dx = 0. Consequently, if the signal is constant or linear over
[a, b)
which
contains
an
interval
[k2−n , 3(k + 1)2−n ) , then the wavelet coefficient
βn
k will equal 0. In Figure 4(a) we show graphs
of the magnitudes of the highest level Haar coefficients and Daub4 coefficients. Each magni−12
tude |β12
k | is plotted at the x -coordinate k2
12
for k = 0, 1, . . . , 2 − 1 . These graphs show that
the highest level Haar coefficients are near 0 over
the constant parts of F , while the highest level
Daub4 coefficients are near 0 over the constant
and linear parts of F . In Figure 4(b) we show
graphs of the sums of the squares of all the coefficients, which show that almost all the Daub4
coefficients are near 0 over the constant and linear parts of F , while the Haar coefficients are
near 0 only over the constant parts of F . Furthermore, the largest magnitude Daub4 coefficients are concentrated around the locations of
the points of nondifferentiability of F . This kind
of local analysis illustrates one of the powerful
features of wavelet analysis.
Looking again at Figure 3(b)[bottom], we see
that the most serious defects of the Daub4 compressed signal are near the points where F is nondifferentiable. If, however, we consider the interval [0.4, 0.6] where F is constant, the Daub4
compressed signal values differ from the values
of F by no more than 1.2 × 10−15 at all of the
1641 discrete values of x in this interval. In
contrast, a Fourier series partial sum using 23
coefficients differs by more than 10−3 at 1441
of these 1641 values of x . The Fourier series partial sum exhibits oscillations of amplitude
6.5 × 10−3 around the value 1 over this subinVOLUME 44, NUMBER 6
walker.qxp 3/24/98 2:18 PM Page 665
Figure 5. (a)[top] Signal. (b)[top] 37-term Daub4 approximation. (a)[bottom] 257-term Fourier cosine
series approximation. (b)[bottom] Highest level Daub4 wavelet coefficients of signal.
terval. Because the Fourier coefficients for F are
only O (n−2 ) , using just the first 23 coefficients
produces an oscillatory approximation to F over
all of [0, 1), including the subinterval [0.4, 0.6].
The highest magnitude wavelet coefficients,
however, are concentrated at the corner points
for F , and their terms affect only a small portion of the partial sum (since their basis functions are compactly supported). Consequently,
the wavelet series provides an extremely close
approximation of F over the subinterval
[0.4, 0.6].
A major defect of the Haar wavelet is its discontinuity. For one thing, it is unsatisfying to use
discontinuous functions to approximate continuous ones. Even with discrete signals there can
be undesirable jumps in Haar series partial sum
values (see Figure 3(a)[bottom]). Therefore, we
want to have a wavelet that is continuous. In the
next section we will describe the Daubechies
wavelets, which have their first L ≥ 2 moments
equal to zero and are continuous.
Daubechies Wavelets
It is possible to generalize the construction of
the Haar wavelet so as to obtain a continuous
scaling function φ(x) and a continuous wavelet
ψ (x) . Moreover, Daubechies has shown how to
make them compactly supported. We will briefly
sketch the main ideas; more details can be found
in [4, 5, 8, 10].
Generalizing from the case of the Haar
wavelets, we require that φ(x) and ψ (x) satisfy
Z∞
(10)
−∞
Z∞
φ(x) dx = 1,
|φ(x)|2 dx = 1,
−∞
Z∞
|ψ (x)|2 dx = 1.
−∞
JUNE/JULY 1997
The MRA axioms tell us that φ(x) must generate a subspace V0 and that V0 ⊂ V1 . Therefore,
(11)
X
φ(x) =
p
ck 2 φ(2x − k)
k∈Z
for some constants {ck } . A wavelet ψ (x) , for
which {ψ (x − k)} spans the wavelet subspace
W0 , can be defined by5
(12)
ψ (x) =
X
p
(−1)k c1−k 2 φ(2x − k).
k∈Z
Equations (11) and (12) generalize the equations
in (2) for the Haar case. The orthogonality of φ
and ψ leads to the following equation
X
(13)
(−1)k c1−k ck = 0.
k∈Z
This equation, and the second equation in (14)
below, imply the orthogonality of the matrices,
WN, used in the fast wavelet transform which we
shall discuss later in this section.
Combining (11) with the first two integrals in
(10), it follows that
(14)
p
ck = 2,
X
k∈Z
X
|ck |2 = 1.
k∈Z
Similarly, assuming that L = 2 , the equations in
(9) combined with (12) imply
(15)
X
(−1)k ck = 0,
k∈Z
X
k(−1)k ck = 0.
k∈Z
5 A simple proof, based on the MRA axioms, that
{ψ (x − k)} spans W0 can be found in [10].
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And, for L > 2 , equations (9) and (12) yield additional equations similar to the ones in (15).
There is a finite set of coefficients that solves
the equations in (14) and (15), namely,
√
√
1+ 3
3+ 3
√ , c1 =
√ ,
c0 =
4 2
4 2
√
√
(16)
3− 3
1− 3
√
√
c2 =
, c3 =
4 2
4 2
R1
α̃kn by α̃kn = 0 f (x)φ̃n,k (x) dx . And, we periodically extend f with period 1 , also denoting this
periodic extension by f . Then, for n ≥ 0 and
k = 0, 1, . . . , 2n − 1 , we have
Z1
X
n
f (x)2n/2
φ 2n (x + j) − k dx
α̃k =
0
(20)
with all other ck = 0 . Using these values of ck ,
the following iterative solution of (11)
(17)
φ0 (x) = 1[0,1) (x),
p
X
φn (x) =
ck 2 φn−1 (2x − k),
k∈Z
for n ≥ 1,
converges to a continuous function φ(x) supported on [0, 3]. It then follows from (12) that
the wavelet ψ (x) is also continuous and compactly supported on [0, 3]. This wavelet ψ we
have been referring to as the Daub4 wavelet.
The set of coefficients {ck } in (16) is the smallest set of coefficients that produce a continuous
compactly supported scaling function. Other
sets of coefficients, related to higher values of
L , are given in [4] and [12].
Once the scaling function φ(x) and the
wavelet function ψ (x) have been found, then
we proceed as we did above in the Haar case. We
define the coefficients {αkn } and {βn
k } by the
equations in (5), where now φ and ψ are the
Daubechies scaling function and wavelet, respectively. The scaling identity (11) and the
wavelet definition (12) yield the following coefficient relations:
X
n+1
αkn =
cm αm+2k
,
(18)
m∈Z
βn
k
=
X
n+1
(−1)m c1−m αm+2k
.
m∈Z
In order to perform calculations in L2 [0, 1) , we
define the periodized wavelet ψ̃n,k and the periodized scaling function φ̃n,k by equations (3)
and (4), only now using the Daubechies wavelet
ψ and scaling function φ in place of the Haar
wavelet and scaling function. Theorem 2 remains valid using these periodic wavelets, but
the proof is more involved (see section 4.5 of [5]
or section 3.11 of [8]). Therefore, for each
f ∈ L2 [0, 1) we can write
n
(19)
f (x) =
α̃00
+
∞ 2X
−1
X
β̃n
k ψ̃n,k (x),
n=0 k=0
R1
β̃n
α̃00 = 0 f (x) dx
where
and
k =
R1
.
We
also
define
the
coefficients
f
(x)
ψ̃
(x)
dx
n,k
0
666
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AMS
=
j∈Z
X Z j+1
f (x − j)2n/2 φ(2n x − k) dx
j∈Z j
= αkn .
n
Similar arguments show that β̃n
k = βk and
n
n
n
n
α̃k+2n = α̃k and β̃k+2n = β̃k for n ≥ 0 and
k = 0, 1, . . . , 2n − 1 . After periodizing (11) and
(12), it follows that
X
n+1
α̃kn =
cm α̃m+2k
,
(21)
m∈Z
β̃n
k =
X
n+1
(−1)m c1−m α̃m+2k
.
m∈Z
Remark. In the section “The Haar Wavelet” we
saw, for the Haar wavelet ψ , that
ψ̃n,k (x) = 2n/2 ψ (2n x − k) for n ≥ 0 and
k = 0, 1, . . . , 2n − 1 . Almost exactly the same result holds for the Daubechies wavelets. For instance, if ψ is the Daub4 wavelet, then ψ is supported on [0, 3]. It follows, for n ≥ 2 , that on the
unit interval ψ̃n,0 is supported on [0, 3 · 2−n ]. On
the
unit
interval,
we
then
have
ψ̃n,k (x) = 2n/2 ψ (2n x − k) for k = 0, 1, . . . , 2n − 3.
Hence, for n ≥ 2 , the periodized Daub4 wavelets
ψ̃n,k are identical in L2 [0, 1) with the wavelet
functions 2n/2 ψ (2n x − k) , except when
k = 2n − 2 and 2n − 1 . Similar results hold for all
the Daubechies wavelets.
We can discretize the series in (19) by analogy
with the Haar series. The coefficient relations in
(21) yield a fast wavelet transform, W , an orthogonal matrix defined by
W = (W2 ⊕ IM−2 )
· · · (PM/2 ⊕ IM/2 ) (WM/2 ⊕ IM/2 ) PM WM
where each matrix WN is an N × N orthogonal
matrix (as follows from (13) and the second equation in (14)). The matrix WN is used to produce
the (N − 1)st level coefficients {α̃kN−1 } and
{β̃N−1
} from the N th level scaling coefficients
k
N
{α̃k } as follows:
h
iT
WN α̃0N , . . . , α̃2NN −1
h
iT
N−1
= α̃0N−1 , β̃N−1
, . . . , α̃2N−1
.
N−1 −1 , β̃2N−1 −1
0
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walker.qxp 3/24/98 2:18 PM Page 667
If we use the coefficients c0 , c1 , c2 , c3 defined
in (16), then for N > 2 , WN has the following
structure :
c +c c +c 2
0
3
1
For N = 2, W2 = c1 +c3 −c0 −c2 . For other
Daubechies wavelets, there are other finite coefficient sequences {ck } , and the matrices WN
are defined similarly. The permutation matrix PN
is the same one that we defined for the Haar
transform and is used to sort the (N − 1)st level
coefficients so that WN−1 can be applied to the
scaling coefficients {α̃kN−1 }.
As initial data for the wavelet transform we
can, as we did for the Haar transform, use discrete data of the form {fj }M−1
j=0 . The equations
in (15) then provide a discrete analog of the zero
moment conditions in (9) [for L = 2]; hence the
wavelet coefficients will be 0 where the data is
linear. In the last section, we saw how this can
produce effective compression of signals when
just the 0th and 1st moments of ψ are 0 .
It is often the case that the initial data are values of a measured signal, i.e. {fk } = {F(xk )} , for
xk = k2−n , where F is a signal obtained from a
measurement process. As shown in the previous
section, we can interpret the behavior of the discrete case based on properties of the function
F . A measured signal
R ∞F is often described by a
convolution: F(x) = −∞ g(t)µ(x − t) dt, where g
is the signal being measured and µ is called the
instrument function. Such convolutions generally
have greater regularity than a typical function
in L2 (R). For instance, if g ∈ L1 (R) and is supported on a finite interval and µ = 1[−r ,r ] for
some positive r , then F is continuous and supported on a finite interval. By a linear change of
variables, we may then assume that F is supported on [0, 1]. The data {F(xk )} then provide
approximations for the highest level scale coefficients {α̃kR } . If we assume that F has period 1 ,
then
α̃kR = 2−R/2
Z∞
−∞
F(x) 2R φ 2R (x − xk ) dx
Z∞
−∞
F(x) 2R φ(2R (x − xk )) dx ≈ F (xk ) .
This approximation will hold for all period 1 continuous functions F and will be more accurate the
larger the value of 2R = M .
The higher the order of a Daubechies wavelet,
the more of its moments are zero. A Daubechies
wavelet of order 2L is defined by 2L nonzero coefficients {ck } , has its first L moments equal to zero,
and is supported on the interval [0, 2L − 1] . Generally speaking, the more moments that are zero, the
more wavelet coefficients that are nearly vanishing
for smooth functions F . This follows from considering Taylor expansions. Suppose F(x) has an L term Taylor expansion about the point xk = k2−n .
That is,
F(x) =
L−1
X
j=0
+
1 (j)
F (xk ) (x − xk )j
j!
1 (L)
F (tx ) (x − xk )L
L!
where tx lies between x and xk . Suppose also that
ψ is supported on [−a, a] and that ψ has its first L
moments equal to 0 and that |F (L) (x)| is bounded
by a constant B on [(k − a)2−n , (k + a)2−n ] . It then
follows that
B
p
|β̃n
k| ≤
L + 1/2 L!
(22)
a
2n
L+1/2
.
This inequality shows why ψ (x) having zero moments produces a large number of small wavelet coefficients. If F has some smoothness on an interval
(c, d), then wavelet coefficients β̃n
k corresponding to
the basis functions ψ (2n (x − xk )) whose supports
are contained in (c, d) will approach 0 rapidly as n
increases to ∞ .
In addition to the Daubechies wavelets, there is
another class of compactly supported wavelets called
coiflets. These wavelets are also constructed using
the method outlined above. A coiflet of order 3L is
defined by 3L nonzero coefficients {ck } and has its
first L moments equal to zero and is supported on
the interval [−L, 2L − 1] . A coiflet of order 3L is distinguished from a Daubechies wavelet of order 2L
in that, in addition to ψ having its first L moments
equal to zero, the scaling function φ for the coiflet
L − 1 moments vanishing. In particular,
also
R ∞ has
j φ(x) dx = 0, for j = 1, . . . , L − 1. For a coiflet of
x
−∞
order 3L , supported on [−a, a] , an argument similar to the one that proves (22) shows that
B
|α̃kR − CM F(xk )| ≤ p
L + 1/2L!
a
2R
L+1/2
.
≈ CM F (xk ) .
√
Here we have replaced 1/ 2R by the scale factor CM and used
JUNE/JULY 1997
This inequality provides a stronger theoretical justification for using the data {CM F(xk )} in place of
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Figure 6. (a)[top] Signal. (b)[top] Signal’s coiflet30 transform, 7th level coefficients lie above the dotted
line. (b)[middle] 7th level coefficients. (a)[bottom] Signal with added noise. (b)[bottom] Noisy signal’s
coiflet30 transform. The horizontal lines are thresholds equal to ±0.15.
the highest level scaling coefficients, beyond
the argument we gave above for Daubechies
wavelets. The construction of coiflets was first
carried out by Daubechies and named after Coifman (who first suggested them).
Compression of Signals
One of the most important applications of
wavelet analysis is to the compression of signals.
As an example, let us use a Daub4 series to compress the signal {fj = F(j/1024)}1023
j=0 where
F(x) = − log |x − 0.2|. See Figure 5(a)[top]. For
this signal, a partial sum containing 99% of the
energy required only 37 coefficients (see Figure
5(b)[top]). It certainly provides a visually acceptable approximation of {fj } . In particular, the
sharp maximum in the signal near x = 0.2 seems
to be reproduced quite well. The compression
ratio is 1024: 37 ≈ 27: 1 , which is an excellent result considering that we also have 99% accuracy. In addition, wavelet analysis has identified the singularities of F . Notice in Figure
5(b)[bottom] the peak in the wavelet coefficients
is near x = 0.2 , where F has a singularity, and
the largest wavelet coefficient is near x = 1 ,
where the periodic extension of F has a jump discontinuity.
Turning to Fourier series, since the even periodic extension is continuous, we used a discrete
Fourier cosine series to compress this signal. In
Figure 5(a)[bottom] we show a 257-term discrete
Fourier cosine series partial sum for {fj } . Even
using seven times as many coefficients as the
wavelet series, the cosine series cannot reproduce the sharp peak in the signal. Better results
could be obtained in this case by either seg668
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menting the interval and performing a cosine expansion on each segment, or by using a smoother
version of the same idea involving local cosine
bases [3, 9, 12, 1, 5].
One way to quantify the accuracy of these
approximations is to use relative R.M.S. differences. Given two sets of data {fj }M−1
j=0 and
{gj }M−1
j=0 , their relative R.M.S. difference,
relative to {fj } , is defined by
v
v
uM−1
uM−1
uX
.u
u
uX
2
D(f , g) = t
|fj − gj | t
|fj |2 .
j=0
j=0
For the example above, if we denote the wavelet
approximation by f w , then D(f , f w ) =
9.8 × 10−3 . For the Fourier cosine series apf c , we have
proximation, call it
c
−2
D(f , f ) = 2.7 × 10 . A rule of thumb for a visually acceptable approximation is to have a relative R.M.S. difference of less than 10−2. The approximations in this example are consistent with
this rule of thumb.
We can also do more localized analysis with
R.M.S. differences. For example, over the subinterval [.075, .325] centered on the singularity
of F , we find that D(f , f w ) = 9.7 × 10−3 and
D(f , f c ) = 3.2 × 10−2. These numbers confirm
our visual impression that the wavelet series
does a better job reproducing the sharp peak in
the signal. Or, using the subinterval [.25, .75], we
get D(f , f w ) = 1.0 × 10−2 and D(f , f c ) =
3.3 × 10−3 , confirming our impression that both
series do an adequate job approximating {fj }
over this subinterval.
VOLUME 44, NUMBER 6
walker.qxp 3/24/98 2:18 PM Page 669
Figure 7. (a)[top] Denoised signal using wavelet analysis. (a)[bottom] Denoised signal using Fourier
analysis. (b)[top] Fourier coefficients of noisy signal and filter function. (b)[middle] Moduli-squared of
Fourier coefficients of original signal. (b)[bottom] Moduli-squared of Fourier coefficients of wavelet
denoised signal.
Although in the examples we have discussed
so far Fourier analysis did not compress the signals very well, we do not wish to create the impression that this will always be true. In fact, if
a signal is composed of relatively few sinusoids,
then Fourier analysis will provide very good
compression. For example, consider the signal
{fj = f (j/1024)}1023
j=0 where f (x) is defined in (1)
with ν = 280 . The Fourier coefficients for f are
graphed in Figure 1(b)[top]. They tend rapidly to
0 away from the frequencies ±280 ; hence the
signal is composed of relatively few sinusoids.
By computing a Fourier series partial sum that
uses only the 122 Fourier coefficients whose
frequencies are within ±30 of ±280 , we obtained a signal g that was visually indistinguishable from the original signal. In fact,
D(f , g) = 5.1 × 10−3 . However, by compressing
{fj } with the largest 122 Daub4 wavelet coefficients, we obtained D(f , f w ) = 2.7 × 10−1 and
the compressed signal f w was only a crude approximation of the original signal. The reason
that compactly supported wavelets perform
poorly in this case is that the large number of
rapid oscillations in the signal produce a correspondingly large number of high magnitude
wavelet coefficients at the highest levels. Consequently, a significant fraction of all the wavelet
coefficients are of high magnitude, so it is not
possible to significantly compress the signal
using compactly supported wavelets. This example illustrates that wavelet analysis is not a
panacea for the problem of signal compression.
In fact, much work has been done in creating
large collections of wavelet bases and Fourier
bases and choosing for each signal a basis which
best compresses it [12, 9, 3, 5].
JUNE/JULY 1997
Signal Denoising
Wavelet analysis can also be used for removing
noise from signals. As an example, we show in
Figure 6(a)[top] a discrete signal {f (j/1024)}1023
j=0
where f (x) is defined by formula (1) with ν = 80.
Each term of the form
2
(5 cos 2π νx)e−640π (x−k/8)
we shall refer to as a blip. Notice that each blip
is concentrated around x = k/8 , since
2
e−640π (x−k/8) rapidly decreases to 0 away from
x = k/8 . This signal can be interpreted as representing the bit sequence 1 0 1 1 0 1 1 . In Figure
6(a)[bottom] we show this signal after it has
been corrupted by adding noise. In Figure
6(b)[top] we show the coiflet30 wavelet coefficients for the original signal. The rationale for
using wavelets to remove the noise is that the
original signal’s wavelet coefficients are closely
correlated with the points near x = k/8 where the
blips are concentrated. To demonstrate this, we
show in Figure 6(b)[middle] a graph of the 7th
level
wavelet
coefficients
{β̃7k }
27 −1
−7
corresponding to the points {k2 }k=0 on the
unit interval. Comparing this to Figure 6(a)[top],
we can see that the positions of this level’s
largest magnitude wavelet coefficients are closely
correlated with the positions of the blips. Similar graphs could also be drawn for other levels,
but the 7th level coefficients have the largest
magnitude. In Figure 6(b)[bottom] we show the
coiflet30 transform of the noisy signal. In spite
of the noise, the 7th level coefficients clearly
stand out, although in a distorted form. By introducing a threshold, in this case 0.15 , we can
retain these 7th level coefficients and remove
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most of the noise. In Figure 7(a)[top] we show
the reconstructed signal obtained by computing
a partial sum using only those coefficients whose
magnitudes do not fall below 0.15 . This reconstruction is not a flawless reproduction of the
original signal, but nevertheless the amount of
noise has been greatly reduced, and the bit sequence 1 0 1 1 0 1 1 can be determined.
In Figure 7(a)[bottom] we show the denoised
signal obtained by filtering the Fourier coefficients of the noisy signal (see Figure 7(b)[top])
using the method of denoising described in the
section “Frequency Information, Denoising”. In
contrast to the wavelet denoising, the Fourier denoising has retained a significant amount of
noise in the spaces between the blips. The source
of this retained noise is that most of the original noise’s Fourier coefficients are of uniform
magnitude distributed across all frequencies.
Consequently, the filter preserves noise coefficients corresponding to frequencies that were
not present in the original signal. These coefficients generate sinusoids that oscillate across the
entire interval [0, 1]. The noise’s wavelet coefficients also have almost uniform magnitude, but
the thresholding process eliminates them all,
except the ones modifying the 7th level coefficients of the original signal. Since these coefficients’ wavelet basis functions are compactly
supported, this causes distortions in the recovered signal that are limited to neighborhoods of
the positions of the 7th level coefficients. Consequently, there is still noise distorting the blips,
but very little noise in between them.
It is also interesting to observe that the
wavelet reconstructed signal and the original
signal have similar frequency content. In Figure
7(b)[middle] and Figure 7(b)[bottom], we have
graphed the moduli-squared of the Fourier coefficients of the original signal and of the wavelet
denoised signal, respectively. These graphs show
that the frequencies of the wavelet reconstruction are, like the frequencies of the original signal, concentrated around ±80 with the highest
magnitude frequencies located precisely at ±80 .
This shows that the coiflet30 wavelet has the ability to extract frequency information. Much work
has been done in refining this ability, including
the development of another class of bases called
wavelet packets [12, 9, 3, 5].
References
[1] P. Auscher, G. Weiss, and M. V. Wickerhauser,
Local sine and cosine basis of Coifman and Meyer
and the construction of smooth wavelets, Wavelets:
A Tutorial in Theory and Applications (C. K. Chui,
ed.), Academic Press, 1992.
[2] W. Briggs and V. E. Henson, The DFT. An
owner’s manual for the Discrete Fourier Transform, SIAM, 1995.
[3] R. Coifman and M. V. Wickerhauser, Wavelets and
adapted waveform analysis, Wavelets: Mathematics and Applications (J. Benedetto and M. Frazier, eds.), CRC Press, 1994.
[4] I. Daubechies, Ten lectures on wavelets, SIAM,
1992.
[5] E. Hernandez and G. Weiss, A first course on
wavelets, CRC Press, 1996.
[6] S. Mallat, Multiresolution approximation and
wavelet orthonormal bases of L2 (R), Trans. Amer.
Math. Soc. 315 (1989), 69–87.
[7] H. Malvar, Signal processing with lapped transforms, Artech House, 1992.
[8] Y. Meyer, Wavelets and operators, Cambridge
Univ. Press, 1992.
[9] ———, Wavelets: Algorithms and applications, SIAM,
1993.
[10] R. Strichartz, How to make wavelets, MAA
Monthly 100, no. 6 (June–July 1993).
[11] J. Walker, Fast Fourier transforms, second edition, CRC Press, 1996.
[12] M. V. Wickerhauser, Adapted wavelet analysis
from theory to software, IEEE and A. K. Peters,
1994.
Conclusion
In this paper we have tried to show how the two
methodologies of Fourier analysis and wavelet
analysis are used for various kinds of work. Of
course, we have only scratched the surface of
both fields. Much more information can be found
in the references and their bibliographies.
670
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VOLUME 44, NUMBER 6
1. Wavelet-based Image Coding: An Overview
23
This spectral density is shown in Figure 10. From the shape of this density
we see that in order to obtain segments in which the spectrum is flat, we
need to partition the spectrum finely at low frequencies, but only coarsely
at high frequencies. The subbands we obtain by this procedure will be approximately vectors of white noise with variances proportional to the power
spectrum over their frequency range. We can use an procedure similar to
that described for the KLT for coding the output. As we will see below,
this particular partition of the spectrum is closely related to the wavelet
transform.
4 Wavelets: A Different Perspective
4.1
Multiresolution Analyses
The discussion so far has been motivated by probabilistic considerations.
We have been assuming our images can be reasonably well-approximated
by Gaussian random vectors with a particular covariance structure. The
use of the wavelet transform in image coding is motivated by a rather
different perspective, that of approximation theory. We assume that our
images are locally smooth functions and can be well-modeled as piecewise
polynomials. Wavelets provide an efficient means for approximating such
functions with a small number of basis elements. This new perspective
provides some valuable insights into the coding process and has motivated
some significant advances.
We motivate the use of the wavelet transform in image coding using the
notion of a multiresolution analysis. Suppose we want to approximate a
continuous-valued square-integrable function f(x) using a discrete set of
values. For example, f(x) might be the brightness of a one-dimensional image. A natural set of values to use to approximate f(x) is a set of regularlyspaced, weighted local averages of f(x) such as might be obtained from the
sensors in a digital camera.
A simple approximation of f(x) based on local averages is a step function
approximation. Let φ(x) be the box function given by φ(x) = 1 for x ∈ [0, 1)
and 0 elsewhere. A step function approximation to f(x) has the form
Af(x) =
fn φ(x − n),
(1.22)
n
where fn is the height of the step in [n, n + 1). A natural value for the
heights fn is simply the average value of f(x) in the interval [n, n + 1).
n+1
This gives fn = n f(x)dx.
We can generalize this approximation procedure to building blocks other
than the box function. Our more generalized approximation will have the
24
Geoffrey M. Davis, Aria Nosratinia
form
Af(x) =
φ̃(x − n), f(x)φ(x − n).
(1.23)
n
Here φ̃(x) is a weight function and φ(x) is an interpolating function chosen so that φ(x), φ̃(x − n) = δ[n]. The restriction on φ(x) ensures that
our approximation will be exact when f(x) is a linear combination of the
functions
φ(x − n). The functions φ(x) and φ̃(x) are normalized so that
|φ(x)|2dx = |φ̃(x)|2 dx = 1. We will further assume that f(x) is periodic
with an integer period so that we only need a finite number of coefficients
to specify the approximation Af(x).
We can vary the resolution of our approximations by dilating and conj
tracting the functions φ(x) and φ̃(x). Let φj (x) = 2 2 φ(2j x) and φ̃j (x) =
j
2 2 φ̃(2j x). We form the approximation Aj f(x) by projecting f(x) onto the
span of the functions {φj (x − 2−j k)}k∈Z , computing
Aj f(x) =
f(x), φ̃j (x − 2−j k)φj (x − 2−j k).
(1.24)
k
Let Vj be the space spanned by the functions {φj (x−2−j k)}. Our resolution
j approximation Aj f is simply a projection (not necessarily an orthogonal
one) of f(x) onto the span of the functions φj (x − 2−j k).
For our box function example, the approximation Aj f(x) corresponds to
an orthogonal projection of f(x) onto the space of step functions with step
width 2−j . Figure 11 shows the difference between the coarse approximation A0 f(x) on the left and the higher resolution approximation A1 f(x) on
the right. Dilating scaling functions give us a way to construct approximations to a given function at various resolutions. An important observation
is that if a given function is sufficiently smooth, the differences between
approximations at successive resolutions will be small.
Constructing our function φ(x) so that approximations at scale j are
special cases of approximations at scale j + 1 will make the analysis of
differences of functions at successive resolutions much easier. The function
φ(x) from our box function example has this property, since step functions
with width 2−j are special cases of step functions with step width 2−j−1 .
For such φ(x)’s the spaces of approximations at successive scales will be
nested, i.e. we have Vj ⊂ Vj+1 .
The observation that the differences Aj+1 f −Aj f will be small for smooth
functions is the motivation for the Laplacian pyramid [26], a way of transforming an image into a set of small coefficients. The 1-D analog of the
procedure is as follows: we start with an initial discrete representation of
a function, the N coefficients of Aj f. We first split this function into the
sum
Aj f(x) = Aj−1 f(x) + [Aj f(x) − Aj−1 f(x)].
(1.25)
Because of the nesting property of the spaces Vj , the difference Aj f(x) −
Aj−1 f(x) can be represented exactly as a sum of N translates of the func-
1. Wavelet-based Image Coding: An Overview
1
1
0.9
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25
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FIGURE 11. A continuous function f (x) (plotted as a dotted line) and box function approximations (solid lines) at two resolutions. On the left is the coarse
approximation A0 f (x) and on the right is the higher resolution approximation
A1 f (x).
tion φj (x). The key point is that the coefficients of these φj translates will
be small provided that f(x) is sufficiently smooth, and hence easy to code.
Moreover, the dimension of the space Vj−1 is only half that of the space
Vj , so we need only N2 coefficients to represent Aj−1 f. (In our box-function
example, the function Aj−1 f is a step function with steps twice as wide as
Aj f, so we need only half as many coefficients to specify Aj−1 f.) We have
partitioned Aj f into N difference coefficients that are easy to code and
N
2 coarse-scale coefficients. We can repeat this process on the coarse-scale
coefficients, obtaining N2 easy-to-code difference coefficients and N4 coarser
scale coefficients, and so on. The end result is 2N − 1 difference coefficients
and a single coarse-scale coefficient.
Burt and Adelson [26] have employed a two-dimensional version of the
above procedure with some success for an image coding scheme. The main
problem with this procedure is that the Laplacian pyramid representation
has more coefficients to code than the original image. In 1-D we have twice
as many coefficients to code, and in 2-D we have 43 as many.
4.2
Wavelets
We can improve on the Laplacian pyramid idea by finding a more efficient
representation of the difference Dj−1 f = Aj f − Aj−1 f. The idea is that
to decompose a space of fine-scale approximations Vj into a direct sum
of two subspaces, a space Vj−1 of coarser-scale approximations and its
complement, Wj−1 . This space Wj−1 is a space of differences between coarse
and fine-scale approximations. In particular, Aj f − Aj−1 f ∈ W j−1 for any
f. Elements of the space can be thought of as the additional details that
must be supplied to generate a finer-scale approximation from a coarse one.
Consider our box-function example. If we limit our attention to functions
on the unit interval, then the space Vj is a space of dimension 2j . We can
26
Geoffrey M. Davis, Aria Nosratinia
0.5
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0
FIGURE 12. D f (x), the difference between the coarse approximation A0 f (x)
and the finer scale approximation A1 f (x) from figure 11.
φ (x)
1
ψ (x)
1
1
1
-1
-1
FIGURE 13. The Haar scaling function and wavelet.
decompose Vj into the space Vj−1 , the space of resolution 2−j+1 approximations, and Wj−1 , the space of details. Because Vj−1 is of dimension 2j−1 ,
Wj−1 must also have dimension 2j−1 for the combined space Vj to have
dimension 2j . This observation about the dimension of Wj provides us with
a means to circumvent the Laplacian pyramid’s problems with expansion.
Recall that in the Laplacian pyramid we represent the difference Dj−1 f
as a sum of N fine-scale basis functions φj (x). This is more information than
we need, however, because the space of functions Dj−1 f is spanned by just
j
N
j
−j
2 basis functions. Let ck be the expansion coefficient φ̃ (x − 2 k), f(x)
in the resolution j approximation to f(x). For our step functions, each
coefficient cj−1
is the average of the coefficients cj2k and cj2k+1 from the
k
resolution j approximation. In order to reconstruct Aj f from Aj−1 f, we
only need the N2 differences cj2k+1 − cj2k . Unlike the Laplacian pyramid,
there is no expansion in the number of coefficients needed if we store these
differences together with the coefficients for Aj−1 f.
The differences cj2k+1 − cj2k in our box function example correspond
(up to a normalizing constant) to coefficients of a basis expansion of the
space of details Wj−1 . Mallat has shown that in general the basis for
Wj consists of translates and dilates of a single prototype function ψ(x),
called a wavelet [27]. The basis for Wj is of the form ψj (x − 2−j k) where
j
ψj (x) = 2 2 ψ(x).
1. Wavelet-based Image Coding: An Overview
27
Figure 13 shows the scaling function (a box function) for our box function example together with the corresponding wavelet, the Haar wavelet.
Figure 12 shows the function D0 f(x), the difference between the approximations A1 f(x) and A1 f(x) from Figure 11. Note that each of the intervals
separated by the dotted lines contains a translated multiple of ψ(x).
The dynamic range of the differences D0 f(x) in Figure 12 is much smaller
than that of A1 f(x). As a result, it is easier to code the expansion coefficients of D0 f(x) than to code those of the higher resolution approximation
A1 f(x). The splitting A1 f(x) into the sum A0 f(x) + D0 f(x) performs a
packing much like that done by the Karhunen-Loève transform. For smooth
functions f(x) the result of the splitting of A1 f(x) into a sum of a coarser
approximation and details is that most of the variation is contained in A0 f,
and D0 f is near zero. By repeating this splitting procedure, partitioning
A0 f(x) into A−1 f(x) + D−1 f(x), we obtain the wavelet transform. The
result is that an initial function approximation Aj f(x) is decomposed into
the telescoping sum
Aj f(x) = Dj−1 f(x) + Dj−2 f(x) + . . . + Dj−n f(x) + Aj−n f(x). (1.26)
The coefficients of the differences Dj−k f(x) are easier to code than the
expansion coefficients of the original approximation Aj f(x), and there is
no expansion of coefficients as in the Laplacian pyramid.
4.3
Recurrence Relations
For the repeated splitting procedure above to be practical, we will need an
efficient algorithm for obtaining the coefficients of the expansions Dj−k f
from the original expansion coefficients for Aj f. A key property of our
scaling functions makes this possible.
One consequence of our partitioning of the space of resolution j approximations, Vj , into a space of resolution j − 1 approximations Vj−1 and
resolution j − 1 details Wj−1 is that the scaling functions φ(x) possess
self-similarity properties. Because Vj−1 ⊂ Vj , we can express the function
φj−1 (x) as a linear combination of the functions φj (x − n). In particular
we have
φ(x) =
hk φ(2x − k).
(1.27)
k
Similarly, we have
φ̃(x) =
h̃k φ̃(2x − k)
k
ψ(x)
=
gk φ(2x − k)
k
ψ̃(x) =
k
g̃k φ̃(2x − k).
(1.28)
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Geoffrey M. Davis, Aria Nosratinia
These recurrence relations provide the link between wavelet transforms
and subband transforms. Combining ( 4.3) and ( 4.3) with ( 4.1), we obtain a simple means for splitting the N expansion coefficients for Aj f into
the N2 expansion coefficients for the coarser-scale approximation Aj−1 f and
the N2 coefficients for the details Dj−1 f. Both the coarser-scale approximation coefficients and the detail coefficients are obtained by convolving the
coefficients of Aj f with a filter and downsampling by a factor of 2. For the
coarser-scale approximation, the filter is a low-pass filter with taps given by
h̃−k . For the details, the filter is a high-pass filter with taps g̃−k . A related
derivation shows that we can invert the split by upsampling the coarserscale approximation coefficients and the detail coefficients by a factor of
2, convolving them with synthesis filters with taps hk and gk , respectively,
and adding them together.
We begin the forward transform with a signal representation in which
we have very fine temporal localization of information but no frequency localization of information. Our filtering procedure splits our signal into lowpass and high-pass components and downsamples each. We obtain twice
the frequency resolution at the expense of half of our temporal resolution.
On each successive step we split the lowest frequency signal component
in to a low pass and high pass component, each time gaining better frequency resolution at the expense of temporal resolution. Figure 14 shows
the partition of the time-frequency plane that results from this iterative
splitting procedure. As we discussed in Section 3.5, such a decomposition,
with its wide subbands in the high frequencies and narrow subbands at low
frequencies leads to effective data compression for a common image model,
a Gaussian random process with an exponentially decaying autocorrelation
function.
The recurrence relations give rise to a fast algorithm for splitting a finescale function approximation into a coarser approximation and a detail
function. If we start with an N coefficient expansion Aj f, the first split
requires kN operations, where k depends on the lengths of the filters we use.
The approximation AJ−1 has N2 coefficients, so the second split requires
Initial time/
frequency
localization
Time/frequency
localization after
first split
Time/frequency
localization after
second split
first high pass
first high pass
frequency
split
split
second high pass
first low pass
second low pass
time
FIGURE 14. Partition of the time-frequency plane created by the wavelet transform.
1. Wavelet-based Image Coding: An Overview
29
k N2 operations. Each successive split requires half as much work, so the
overall transform requires O(N ) work.
4.4
Wavelet Transforms vs. Subband Decompositions
The wavelet transform is a special case of a subband transform, as the
derivation of the fast wavelet transform reveals. What, then, does the
wavelet transform contribute to image coding? As we discuss below, the
chief contribution of the wavelet transform is one of perspective. The mathematical machinery used to develop the wavelet transform is quite different
than that used for developing subband coders. Wavelets involve the analysis of continuous functions whereas analysis of subband decompositions is
more focused on discrete time signals. The theory of wavelets has a strong
spatial component whereas subbands are more focused in the frequency
domain.
The subband and wavelet perspectives represent two extreme points in
the analysis of this iterated filtering and downsampling process. The filters
used in subband decompositions are typically designed to optimize the
frequency domain behavior of a single filtering and subsampling. Because
wavelet transforms involve iterated filtering and downsampling, the analysis
of a single iteration is not quite what we want. The wavelet basis functions
can be obtained by iterating the filtering and downsampling procedure an
infinite number of times. Although in applications we iterate the filtering
and downsampling procedure only a small number of times, examination
of the properties of the basis functions provides considerable insight into
the effects of iterated filtering.
A subtle but important point is that when we use the wavelet machinery, we are implicitly assuming that the values we transform are actually
fine-scale scaling function coefficients rather than samples of some function.
Unlike the subband framework, the wavelet framework explicitly specifies
an underlying continuous-valued function from which our initial coefficients
are derived. The use of continuous-valued functions allows the use of powerful analytical tools, and it leads to a number of insights that can be used
to guide the filter design process. Within the continuous-valued framework
we can characterize the types of functions that can be represented exactly
with a limited number of wavelet coefficients. We can also address issues
such as the smoothness of the basis functions. Examination of these issues has led to important new design criteria for both wavelet filters and
subband decompositions.
A second important feature of the wavelet machinery is that it involves
both spatial as well as frequency considerations. The analysis of subband
decompositions is typically more focused on the frequency domain. Coefficients in the wavelet transform correspond to features in the underlying
function in specific, well-defined locations. As we will see below, this explicit use of spatial information has proven quite valuable in motivating
30
Geoffrey M. Davis, Aria Nosratinia
some of the most effective wavelet coders.
4.5
Wavelet Properties
There is an extensive literature on wavelets and their properties. See [28], [23],
or [29] for an introduction. Properties of particular interest for image compression are the the accuracy of approximation , the smoothness, and the
support of these bases.
The functions φ(x) and ψ(x) are the building blocks from which we construct our compressed images. When compressing natural images, which
tend contain locally smooth regions, it is important that these building
blocks be reasonably smooth. If the wavelets possess discontinuities or
strong singularities, coefficient quantization errors will cause these discontinuities and singularities to appear in decoded images. Such artifacts
are highly visually objectionable, particularly in smooth regions of images.
Procedures for estimating the smoothness of wavelet bases can be found
in [30] and [31]. Rioul [32] has found that under certain conditions that
the smoothness of scaling functions is a more important criterion than a
standard frequency selectivity criterion used in subband coding.
Accuracy of approximation is a second important design criterion that
has arisen from wavelet framework. A remarkable fact about wavelets is
that it is possible to construct smooth, compactly supported bases that can
exactly reproduce any polynomial up to a given degree. If a continuousvalued function f(x) is locally equal to a polynomial, we can reproduce that
portion of f(x) exactly with just a few wavelet coefficients. The degree
of the polynomials that can be reproduced exactly is determined by the
number of vanishing moments of the dual wavelet
ψ̃(x). The dual wavelet
ψ̃(x) has N vanishing moments provided that xk ψ̃(x)dx = 0 for k =
0, . . . , N . Compactly supported bases for L2 for which ψ̃(x) has N vanishing
moments can locally reproduce polynomials of degree N − 1.
The number of vanishing moments also determines the rate of convergence of the approximations Aj f to the original function f as the resolution
goes to infinity. It has been shown that f −Aj f
≤ C2−jN f (N) where N
is the number of vanishing moments of ψ̃(x) and f (N) is the N th derivative
of f [33, 34, 35].
The size of the support of the wavelet basis is another important design criterion. Suppose that the function f(x) we are transforming is equal
to polynomial of degree N − 1 in some region. If ψ̃ has has N vanishing
moments, then any basis function for which the corresponding dual function lies entirely in the region in which f is polynomial will have a zero
coefficient. The smaller the support of ψ̃ is, the more zero coefficients we
will obtain. More importantly, edges produce large wavelet coefficients. The
larger ψ̃ is, the more likely it is to overlap an edge. Hence it is important
that our wavelets have reasonably small support.
1. Wavelet-based Image Coding: An Overview
31
There is a tradeoff between wavelet support and the regularity and accuracy of approximation. Wavelets with short support have strong constraints
on their regularity and accuracy of approximation, but as the support is
increased they can be made to have arbitrary degrees of smoothness and
numbers of vanishing moments. This limitation on support is equivalent to
keeping the analysis filters short. Limiting filter length is also an important
consideration in the subband coding literature, because long filters lead to
ringing artifacts around edges.
5 A Basic Wavelet Image Coder
State-of-the-art wavelet coders are all derived from the transform coder
paradigm. There are three basic components that underly current wavelet
coders: a decorrelating transform, a quantization procedure, and an entropy
coding procedure. Considerable current research is being performed on all
three of these components. Before we discuss state-of-the-art coders in the
next sections, we will describe a basic wavelet transform coder and discuss
optimized versions of each of the components.7
5.1
Choice of Wavelet Basis
Deciding on the optimal wavelet basis to use for image coding is a difficult
problem. A number of design criteria, including smoothness, accuracy of
approximation, size of support, and filter frequency selectivity are known
to be important. However, the best combination of these features is not
known.
The simplest form of wavelet basis for images is a separable basis formed
from translations and dilations of products of one dimensional wavelets. Using separable transforms reduces the problem of designing efficient wavelets
to a one-dimensional problem, and almost all current coders employ separable transforms. Recent work of Sweldens and Kovačević [36] simplifies
considerably the design of non-separable bases, and such bases may prove
more efficient than separable transforms.
The prototype basis functions for separable transforms are φ(x)φ(y),
φ(x)ψ(y), ψ(x)φ(y), and ψ(x)ψ(y). Each step of the transform for such
bases involves two frequency splits instead of one. Suppose we have an
N × N image. First each of the N rows in the image is split into a lowpass half and a high pass half. The result is an N × N2 sub-image and an
N × N2 high-pass sub-image. Next each column of the sub-images is split
into a low-pass and a high-pass half. The result is a four-way partition
7 C++ source code for a coder that implements these components is available from
the web site http://www.cs.dartmouth.edu/∼gdavis/wavelet/wavelet.html.
32
Geoffrey M. Davis, Aria Nosratinia
of the image into horizontal low-pass/vertical low-pass, horizontal highpass/vertical low-pass, horizontal low-pass/vertical high-pass, and horizontal high-pass/vertical high-pass sub-images. The low-pass/low-pass subimage is subdivided in the same manner in the next step as is illustrated
in Figure 17.
Unser [35] shows that spline wavelets are attractive for coding applications based on approximation theoretic considerations. Experiments by
Rioul [32] for orthogonal bases indicate that smoothness is an important
consideration for compression. Experiments by Antonini et al [37] find that
both vanishing moments and smoothness are important, and for the filters
tested they found that smoothness appeared to be slightly more important than the number of vanishing moments. Nonetheless, Vetterli and
Herley [38] state that “the importance of regularity for signal processing
applications is still an open question.” The bases most commonly used
in practice have between one and two continuous derivatives. Additional
smoothness does not appear to yield significant improvements in coding
results.
Villasenor et al [39] have systematically examined all minimum order
biorthogonal filter banks with lengths ≤ 36. In addition to the criteria
already mentioned, [39] also examines measures of oscillatory behavior and
of the sensitivity of the coarse-scale approximations Aj f(x) to translations
of the function f(x). The best filter found in these experiments was a 7/9tap spline variant with less dissimilar lengths from [37], and this filter is
one of the most commonly used in wavelet coders.
There is one caveat with regard to the results of the filter evaluation
in [39]. Villasenor et al compare peak signal to noise ratios generated by a
simple transform coding scheme. The bit allocation scheme they use works
well for orthogonal bases, but it can be improved upon considerably in
the biorthogonal case. This inefficient bit allocation causes some promising
biorthogonal filter sets to be overlooked.
For biorthogonal transforms, the squared error in the transform domain
is not the same as the squared error in the original image. As a result,
the problem of minimizing image error is considerably more difficult than
in the orthogonal case. We can reduce image-domain errors by performing
bit allocation using a weighted transform-domain error measure that we
discuss in section 5.5. A number of other filters yield performance comparable to that of the 7/9 filter of [37] provided that we do bit allocation
with a weighted error measure. One such basis is the Deslauriers-Dubuc
interpolating wavelet of order 4 [40, 41], which has the advantage of having
filter taps that are dyadic rationals. Both the spline wavelet of [37] and the
order 4 Deslauriers-Dubuc wavelet have 4 vanishing moments in both ψ(x)
andψ̃(x), and the basis functions have just under 2 continuous derivatives
in the L2 sense.
One new very promising set of filters has been developed by Balasingham
and Ramstad [42]. Their design procedure combines classical filter design
1. Wavelet-based Image Coding: An Overview
33
Dead zone
x
x=0
FIGURE 15. Dead-zone quantizer, with larger encoder partition around x = 0
(dead zone) and uniform quantization elsewhere.
techniques with ideas from wavelet constructions and yields filters that
perform significantly better than the popular 7/9 filter set from [37].
5.2
Boundaries
Careful handling of image boundaries when performing the wavelet transform is essential for effective compression algorithms. Naive techniques for
artificially extending images beyond given boundaries such as periodization
or zero-padding lead to significant coding inefficiencies. For symmetrical
wavelets an effective strategy for handling boundaries is to extend the image via reflection. Such an extension preserves continuity at the boundaries
and usually leads to much smaller wavelet coefficients than if discontinuities
were present at the boundaries. Brislawn [43] describes in detail procedures
for non-expansive symmetric extensions of boundaries. An alternative approach is to modify the filter near the boundary. Boundary filters [44, 45]
can be constructed that preserve filter orthogonality at boundaries. The
lifting scheme [46] provides a related method for handling filtering near the
boundaries.
5.3
Quantization
Most current wavelet coders employ scalar quantization for coding. There
are two basic strategies for performing the scalar quantization stage. If
we knew the distribution of coefficients for each subband in advance, the
optimal strategy would be to use entropy-constrained Lloyd-Max quantizers
for each subband. In general we do not have such knowledge, but we can
provide a parametric description of coefficient distributions by sending side
information. Coefficients in the high pass subbands of a wavelet transform
are known a priori to be distributed as generalized Gaussians [27] centered
around zero.
A much simpler quantizer that is commonly employed in practice is a
uniform quantizer with a dead zone. The quantization bins, as shown in
Figure 15, are of the form [n∆, (n + 1)∆) for n ∈ Z except for the central
bin [−∆, ∆). Each bin is decoded to the value at its center in the simplest
case, or to the centroid of the bin. In the case of asymptotically high rates,
uniform quantization is optimal [47]. Although in practical regimes these
dead-zone quantizers are suboptimal, they work almost as well as Lloyd-
34
Geoffrey M. Davis, Aria Nosratinia
Max coders when we decode to the bin centroids [48]. Moreover, dead-zone
quantizers have the advantage that of being very low complexity and robust
to changes in the distribution of coefficients in source. An additional advantage of these dead-zone quantizers is that they can be nested to produce
an embedded bitstream following a procedure in [49].
5.4
Entropy Coding
Arithmetic coding provides a near-optimal entropy coding for the quantized coefficient values. The coder requires an estimate of the distribution
of quantized coefficients. This estimate can be approximately specified by
providing parameters for a generalized Gaussian or a Laplacian density.
Alternatively the probabilities can be estimated online. Online adaptive
estimation has the advantage of allowing coders to exploit local changes
in image statistics. Efficient adaptive estimation procedures are discussed
in [50] and [51].
Because images are not jointly Gaussian random processes, the transform
coefficients, although decorrelated, still contain considerable structure. The
entropy coder can take advantage of some of this structure by conditioning
the encodings on previously encoded values. A coder of [49] obtains modest
performance improvements using such a technique.
5.5
Bit Allocation
The final question we need to address is that of how finely to quantize each
subband. As we discussed in Section 3.2, the general idea is to determine
the number
of bits bj to devote to coding subband j so that thetotal distortion j Dj (bj ) is minimized subject to the constraint that j bj ≤ b.
Here Dj (b) is the amount of distortion incurred in coding subband j with
b bits. When the functions Dj (b) are known in closed form we can solve
the problem using the Kuhn-Tucker conditions. One common practice is
to approximate the functions Dj (b) with the rate-distortion function for a
Gaussian random variable. However, this approximation is not very accurate at low bit rates. Better results may be obtained by measuring Dj (b)
for a range of values of b and then solving the constrained minimization
problem using integer programming techniques. An algorithm of Shoham
and Gersho [52] solves precisely this problem.
For biorthogonal wavelets we have the additional problem that squared
error in the transform domain is not equal to squared error in the inverted
image. Moulin [53] has formulated a multiscale relaxation algorithm which
provides an approximate solution to the allocation problem for this case.
Moulin’s algorithm yields substantially better results than the naive approach of minimizing squared error in the transform domain.
A simpler approach is to approximate the squared error in the image by
weighting the squared errors in each subband. The weight wj for subband
1. Wavelet-based Image Coding: An Overview
35
j is obtained as follows: we set a single coefficient in subband j to 1 and
set all other wavelet coefficients to zero. We then invert this transform.
The weight wj is equal to the sum of the squares of the values in the resulting
inverse transform. We allocate
bits by minimizing the weighted sum
w
D
(b
)
rather
than
the
sum
j
j
j
j
j Dj (bj ). Further details may be found
in Naveen and Woods [54]. This weighting procedure results in substantial
coding improvements when using wavelets that are not very close to being orthogonal, such as the Deslauriers-Dubuc wavelets popularized by the
lifting scheme [46]. The 7/9 tap filter set of [37], on the other hand, has
weights that are all nearly 1, so this weighting provides little benefit.
5.6
Perceptually Weighted Error Measures
Our goal in lossy image coding is to minimize visual discrepancies between
the original and compressed images. Measuring visual discrepancy is a difficult task. There has been a great deal of research on this problem, but
because of the great complexity of the human visual system, no simple,
accurate, and mathematically tractable measure has been found.
Our discussion up to this point has focused on minimizing squared error distortion in compressed images primarily because this error metric is
mathematically convenient. The measure suffers from a number of deficits,
however. For example, consider two images that are the same everywhere
except in a small region. Even if the difference in this small region is large
and highly visible, the mean squared error for the whole image will be small
because the discrepancy is confined to a small region. Similarly, errors that
are localized in straight lines, such as the blocking artifacts produced by
the discrete cosine transform, are much more visually objectionable than
squared error considerations alone indicate.
There is evidence that the human visual system makes use of a multiresolution image representation; see [55] for an overview. The eye is much more
sensitive to errors in low frequencies than in high. As a result, we can improve the correspondence between our squared error metric and perceived
error by weighting the errors in different subbands according to the eye’s
contrast sensitivity in a corresponding frequency range. Weights for the
commonly used 7/9-tap filter set of [37] have been computed by Watson et
al in [56].
6 Extending the Transform Coder Paradigm
The basic wavelet coder discussed in Section 5 is based on the basic transform coding paradigm, namely decorrelation and compaction of energy into
a few coefficients. The mathematical framework used in deriving the wavelet
transform motivates compression algorithms that go beyond the traditional
36
Geoffrey M. Davis, Aria Nosratinia
mechanisms used in transform coding. These important extensions are at
the heart of modern wavelet coding algorithms of Sections 7 and 9. We take
a moment here to discuss these extensions.
Conventional transform coding relies on energy compaction in an ordered
set of transform coefficients, and quantizes those coefficients with a priority
according to their order. This paradigm, while quite powerful, is based on
several assumptions about images that are not always completely accurate.
In particular, the Gaussian assumption breaks down for the joint distributions across image discontinuities. Mallat and Falzon [57] give the following
example of how the Gaussian, high-rate analysis breaks down at low rates
for non-Gaussian processes.
Let Y [n] be a random N -vector defined by

 X if n = P
Y [n] =
X if n = P + 1(modN )
(1.29)

0 otherwise
Here P is a random integer uniformly distributed between 0 and N − 1 and
X is a random variable that equals 1 or -1 each with probability 12 . X and
P are independent. The vector Y has zero mean and a covariance matrix
with entries
 2
 N for n = m
1
E{Y [n]Y [m]} =
for |n − m| ∈ {1, N − 1}
(1.30)
 N
0 otherwise
The covariance matrix is circulant, so the KLT for this process is the simply
the Fourier transform. The Fourier transform of Y is a very inefficient reprek
sentation for coding Y . The energy at frequency k will be |1+e2πi N |2 which
means that the energy of Y is spread out over the entire low-frequency half
of the Fourier basis with some spill-over into the high-frequency half. The
KLT has “packed” the energy of the two non-zero coefficients of Y into
roughly N2 coefficients. It is obvious that Y was much more compact in its
original form, and could be coded better without transformation: Only two
coefficients in Y are non-zero, and we need only specify the values of these
coefficients and their positions.
As suggested by the example above, the essence of the extensions to
traditional transform coding is the idea of selection operators. Instead of
quantizing the transform coefficients in a pre-determined order of priority, the wavelet framework lends itself to improvements, through judicious
choice of which elements to code. This is made possible primarily because
wavelet basis elements are spatially as well as spectrally compact. In parts
of the image where the energy is spatially but not spectrally compact (like
the example above) one can use selection operators to choose subsets of
the wavelet coefficients that represent that signal efficiently. A most notable example is the Zerotree coder and its variants (Section 7).
1. Wavelet-based Image Coding: An Overview
37
More formally, the extension consists of dropping the constraint of linear image approximations, as the selection operator is nonlinear. The work
of DeVore et al. [58] and of Mallat and Falzon [57] suggests that at low
rates, the problem of image coding can be more effectively addressed as a
problem in obtaining a non-linear image approximation. This idea leads to
some important differences in coder implementation compared to the linear
framework. For linear approximations, Theorems 3.1 and 3.3 in Section 3.1
suggest that at low rates we should approximate our images using a fixed
subset of the Karhunen-Loève basis vectors. We set a fixed set of transform
coefficients to zero, namely the coefficients corresponding to the smallest
eigenvalues of the covariance matrix. The non-linear approximation idea,
on the other hand, is to approximate images using a subset of basis functions that are selected adaptively based on the given image. Information
describing the particular set of basis functions used for the approximation,
called a significance map, is sent as side information. In Section 7 we describe zerotrees, a very important data structure used to efficiently encode
significance maps.
Our example suggests that a second important assumption to relax is
that our images come from a single jointly Gaussian source. We can obtain
better energy packing by optimizing our transform to the particular image
at hand rather than to the global ensemble of images. The KLT provides
efficient variance packing for vectors drawn from a single Gaussian source.
However, if we have a mixture of sources the KLT is considerably less efficient. Frequency-adaptive and space/frequency-adaptive coders decompose
images over a large library of different bases and choose an energy-packing
transform that is adapted to the image itself. We describe these adaptive
coders in Section 8.
Trellis coded quantization represents a more drastic departure from the
transform coder framework. While TCQ coders operate in the transform
domain, they effectively do not use scalar quantization. Trellis coded quantization captures not only correlation gain and fractional bitrates, but also
the packing gain of VQ. In both performance and complexity, TCQ is essentially VQ in disguise.
The selection operator that characterizes the extension to the transform
coder paradigm generates information that needs to be conveyed to the
decoder as “side information”. This side information can be in the form
of zerotrees, or more generally energy classes. Backward mixture estimation represents a different approach: it assumes that the side information
is largely redundant and can be estimated from the causal data. By cutting down on the transmitted side information, these algorithms achieve a
remarkable degree of performance and efficiency.
For reference, Table 1.1 provides a comparison of the peak signal to
38
Geoffrey M. Davis, Aria Nosratinia
TABLE 1.1. Peak signal to noise ratios in decibels for coders discussed
in the paper. Higher values indicate better performance.
Lena (bits/pixel)
Barbara (bits/pixel)
Type of Coder
1.0
0.5
0.25
1.0
0.5
0.25
JPEG [59]
37.9 34.9 31.6 33.1 28.3
25.2
Optimized JPEG [60]
39.6 35.9 32.3 35.9 30.6
26.7
Baseline Wavelet [61]
39.4 36.2 33.2 34.6 29.5
26.6
Zerotree (Shapiro) [62]
39.6 36.3 33.2 35.1 30.5
26.8
Zerotree (Said & Pearlman) [63]
40.5 37.2 34.1 36.9 31.7
27.8
Zerotree (R/D optimized) [64]
40.5 37.4 34.3 37.0 31.3
27.2
Frequency-adaptive [65]
39.3 36.4 33.4 36.4 31.8
28.2
Space-frequency adaptive [66]
40.1 36.9 33.8 37.0 32.3
28.7
Frequency-adaptive + zerotrees [67]
40.6 37.4 34.4 37.7 33.1
29.3
TCQ subband [68]
41.1 37.7 34.3
–
–
–
Bkwd. mixture estimation (EQ) [69] 40.9 37.7 34.6
–
–
–
noise ratios for the coders we discuss in the paper.8 The test images are
the 512×512 Lena image and the 512×512 Barbara image. Figure 16 shows
the Barbara image as compressed by JPEG, a baseline wavelet transform
coder, and the zerotree coder of Said and Pearlman [63]. The Barbara image
is particularly difficult to code, and we have compressed the image at a low
rate to emphasize coder errors. The blocking artifacts produced by the discrete cosine transform are highly visible in the image on the top right. The
difference between the two wavelet coded images is more subtle but quite
visible at close range. Because of the more efficient coefficient encoding (to
be discussed below), the zerotree-coded image has much sharper edges and
better preserves the striped texture than does the baseline transform coder.
7 Zerotree Coding
The rate-distortion analysis of the previous sections showed that optimal
bitrate allocation is achieved when the signal is divided into subbands such
that each subband contains a “white” signal. It was also shown that for
typical signals of interest, this leads to narrower bands in the low frequencies and wider bands in the high frequencies. Hence, wavelet transforms
have very good energy compaction properties.
This energy compaction leads to efficient utilization of scalar quantizers.
However, a cursory examination of the transform in Figure 17 shows that
a significant amount of structure is present, particularly in the fine scale
coefficients. Wherever there is structure, there is room for compression, and
8 More
current numbers may be found on the web at
http://www.icsl.ucla.edu/∼ipl/psnr results.html
1. Wavelet-based Image Coding: An Overview
39
FIGURE 16. Results of different compression schemes for the 512 × 512 Barbara
test image at 0.25 bits per pixel. Top left: original image. Top right: baseline
JPEG, PSNR = 24.4 dB. Bottom left: baseline wavelet transform coder [61],
PSNR = 26.6 dB. Bottom right: Said and Pearlman zerotree coder, PSNR =
27.6 dB.
40
Geoffrey M. Davis, Aria Nosratinia
advanced wavelet compression algorithms all address this structure in the
higher frequency subbands.
One of the most prevalent approaches to this problem is based on exploiting the relationships of the wavelet coefficients across bands. A direct
visual inspection indicates that large areas in the high frequency bands have
little or no energy, and the small areas that have significant energy are similar in shape and location, across different bands. These high-energy areas
stem from poor energy compaction close to the edges of the original image.
Flat and slowly varying regions in the original image are well-described by
the low-frequency basis elements of the wavelet transform (hence leading to
high energy compaction). At the edge locations, however, low-frequency basis elements cannot describe the signal adequately, and some of the energy
leaks into high-frequency coefficients. This happens similarly at all scales,
thus the high-energy high-frequency coefficients representing the edges in
the image have the same shape.
Our a priori knowledge that images of interest are formed mainly from
flat areas, textures, and edges, allows us to take advantage of the resulting
cross-band structure. Zerotree coders combine the idea of cross-band correlation with the notion of coding zeros jointly (which we saw previously
in the case of JPEG), to generate very powerful compression algorithms.
The first instance of the implementation of zerotrees is due to Lewis
and Knowles [70]. In their algorithm the image is represented by a treestructured data construct (Figure 18). This data structure is implied by a
dyadic discrete wavelet transform (Figure 19) in two dimensions. The root
node of the tree represents the scaling function coefficient in the lowest
frequency band, which is the parent of three nodes. Nodes inside the tree
correspond to wavelet coefficients at a scale determined by their height
in the tree. Each of these coefficients has four children, which correspond
to the wavelets at the next finer scale having the same location in space.
These four coefficients represent the four phases of the higher resolution
basis elements at that location. At the bottom of the data structure lie the
leaf nodes, which have no children.
Note that there exist three such quadtrees for each coefficient in the low
frequency band. Each of these three trees corresponds to one of three filtering orderings: there is one tree consisting entirely of coefficients arising from
horizontal high-pass, vertical low-pass operation (HL); one for horizontal
low-pass, vertical high-pass (LH), and one for high-pass in both directions
(HH).
The zerotree quantization model used by Lewis and Knowles was arrived
at by observing that often when a wavelet coefficient is small, its children
on the wavelet tree are also small. This phenomenon happens because significant coefficients arise from edges and texture, which are local. It is not
difficult to see that this is a form of conditioning. Lewis and Knowles took
this conditioning to the limit, and assumed that insignificant parent nodes
always imply insignificant child nodes. A tree or subtree that contains (or
1. Wavelet-based Image Coding: An Overview
FIGURE 17. Wavelet transform of the image “Lena.”
LL3
LH3
HL3
HH3
LH2
LH1
HH2
HL2
HH1
HL1
Typical
Wavelet tree
FIGURE 18. Space-frequency structure of wavelet transform
41
42
Geoffrey M. Davis, Aria Nosratinia
H
2
H
2
0
H
0
Input
Signal
H
0
1
1
H
1
2
2
H
Low frequency
Component
2
Detail
Components
2
FIGURE 19. Filter bank implementing a discrete wavelet transform
is assumed to contain) only insignificant coefficients is known as a zerotree.
Lewis and Knowles used the following algorithm for the quantization
of wavelet coefficients: Quantize each node according to an optimal scalar
quantizer for the Laplacian density. If the node value is insignificant according to a pre-specified threshold, ignore all its children. These ignored
coefficients will be decoded as zeros at the decoder. Otherwise, go to each
of its four children and repeat the process. If the node was a leaf node and
did not have a child, go to the next root node and repeat the process.
Aside from the nice energy compaction properties of the wavelet transform, the Lewis and Knowles coder achieves its compression ratios by joint
coding of zeros. For efficient run-length coding, one needs to first find a
conducive data structure, e.g. the zig-zag scan in JPEG. Perhaps the most
significant contribution of this work was to realize that wavelet domain
data provide an excellent context for run-length coding: not only are large
run lengths of zeros generated, but also there is no need to transmit the
length of zero runs, because they are assumed to automatically terminate
at the leaf nodes of the tree. Much the same as in JPEG, this is a form of
joint vector/scalar quantization. Each individual (significant) coefficient is
quantized separately, but the symbols corresponding to small coefficients
in fact are representing a vector consisting of that element and the zero
run that follows it to the bottom of the tree.
While this compression algorithm generates subjectively acceptable images, its rate-distortion performance falls short of baseline JPEG, which at
the time was often used for comparison purposes. The lack of sophistication
in the entropy coding of quantized coefficients somewhat disadvantages this
coder, but the main reason for its mediocre performance is the way it generates and recognizes zerotrees. As we have noted, whenever a coefficient
is small, it is likely that its descendents are also insignificant. However, the
Lewis and Knowles algorithm assumes that small parents always have small
descendents, and therefore suffers large distortions when this does not hold
because it zeros out large coefficients. The advantage of this method is that
the detection of zerotrees is automatic: zerotrees are determined by measuring the magnitude of known coefficients. No side information is required
to specify the locations of zerotrees, but this simplicity is obtained at the
cost of reduced performance. More detailed analysis of this tradeoff gave
1. Wavelet-based Image Coding: An Overview
43
rise to the next generation of zerotree coders.
7.1
The Shapiro and Said-Pearlman Coders
The Lewis and Knowles algorithm, while capturing the basic ideas inherent
in many of the later coders, was incomplete. It had all the intuition that
lies at the heart of more advanced zerotree coders, but did not efficiently
specify significance maps, which is crucial to the performance of wavelet
coders.
A significance map is a binary function whose value determines whether
each coefficient is significant or not. If not significant, a coefficient is assumed to quantize to zero. Hence a decoder that knows the significance map
needs no further information about that coefficient. Otherwise, the coefficient is quantized to a non-zero value. The method of Lewis and Knowles
does not generate a significance map from the actual data, but uses one
implicitly, based on a priori assumptions on the structure of the data. On
the infrequent occasions when this assumption does not hold, a high price
is paid in terms of distortion. The methods to be discussed below make use
of the fact that, by using a small number of bits to correct mistakes in our
assumptions about the occurrences of zerotrees, we can reduce the coded
image distortion considerably.
The first algorithm of this family is due to Shapiro [71] and is known
as the embedded zerotree wavelet (EZW) algorithm. Shapiro’s coder was
based on transmitting both the non-zero data and a significance map. The
bits needed to specify a significance map can easily dominate the coder
output, especially at lower bitrates. However, there is a great deal of redundancy in a general significance map for visual data, and the bitrates
for its representation can be kept in check by conditioning the map values
at each node of the tree on the corresponding value at the parent node.
Whenever an insignificant parent node is observed, it is highly likely that
the descendents are also insignificant. Therefore, most of the time, a “zerotree” significance map symbol is generated. But because p, the probability
of this event, is close to 1, its information content, −p log p, is very small.
So most of the time, a very small amount of information is transmitted,
and this keeps the average bitrate needed for the significance map relatively
small.
Once in a while, one or more of the children of an insignificant node will
be significant. In that case, a symbol for “isolated zero” is transmitted.
The likelihood of this event is lower, and thus the bitrate for conveying
this information is higher. But it is essential to pay this price to avoid
losing significant information down the tree and therefore generating large
distortions.
In summary, the Shapiro algorithm uses three symbols for significance
maps: zerotree, isolated zero, or significant value. But using this structure,
and by conditionally entropy coding these symbols, the coder achieves very
Geoffrey M. Davis, Aria Nosratinia
0
0
0
0
0
0
0
x
x
1
1
x
x
x
x
x
x
x
x
x
x
0
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
....
x
1
1
1
1
0
0
1
0
0
0
0
0
....
x
x
x
x
x
1
1
1
1
1
1
0
0
....
......
x
0
....
0
......
0
......
0
......
0
......
1
......
1
......
LSB
1
......
MSB
......
44
Wavelet Coefficients
in scan order
FIGURE 20. Bit plane profile for raster scan ordered wavelet coefficients.
good rate-distortion performance.
In addition, Shapiro’s coder also generates an embedded code. Coders
that generate embedded codes are said to have the progressive transmission
or successive refinement property. Successive refinement consists of first
approximating the image with a few bits of data, and then improving the
approximation as more and more information is supplied. An embedded
code has the property that for two given rates R1 > R2 , the rate-R2 code
is a prefix to the rate-R1 code. Such codes are of great practical interest
for the following reasons:
• The encoder can easily achieve a precise bitrate by continuing to
output bits when it reaches the desired rate.
• The decoder can cease decoding at any given point, generating an
image that is the best representation possible with the decoded number of bits. This is of practical interest for broadcast applications
where multiple decoders with varying computational, display, and
bandwidth capabilities attempt to receive the same bitstream. With
an embedded code, each receiver can decode the passing bitstream
according to its particular needs and capabilities.
• Embedded codes are also very useful for indexing and browsing, where
only a rough approximation is sufficient for deciding whether the
image needs to be decoded or received in full. The process of screening
images can be speeded up considerably by using embedded codes:
after decoding only a small portion of the code, one knows if the
target image is present. If not, decoding is aborted and the next image
is requested, making it possible to screen a large number of images
quickly. Once the desired image is located, the complete image is
decoded.
Shapiro’s method generates an embedded code by using a bit-slice approach (see Figure 20). First, the wavelet coefficients of the image are
1. Wavelet-based Image Coding: An Overview
45
indexed into a one-dimensional array, according to their order of importance. This order places lower frequency bands before higher frequency
bands since they have more energy, and coefficients within each band appear in a raster scan order. The bit-slice code is generated by scanning
this one-dimensional array, comparing each coefficient with a threshold T .
This initial scan provides the decoder with sufficient information to recover
the most significant bit slice. In the next pass, our information about each
coefficient is refined to a resolution of T /2, and the pass generates another
bit slice of information. This process is repeated until there are no more
slices to code.
Figure 20 shows that the upper bit slices contain a great many zeros
because there are many coefficients below the threshold. The role of zerotree
coding is to avoid transmitting all these zeros. Once a zerotree symbol
is transmitted, we know that all the descendent coefficients are zero, so
no information is transmitted for them. In effect, zerotrees are a clever
form of run-length coding, where the coefficients are ordered in a way to
generate longer run lengths (more efficient) as well as making the runs
self-terminating, so the length of the runs need not be transmitted.
The zerotree symbols (with high probability and small code length) can
be transmitted again and again for a given coefficient, until it rises above the
sinking threshold, at which point it will be tagged as a significant coefficient.
After this point, no more zerotree information will be transmitted for this
coefficient.
To achieve embeddedness, Shapiro uses a clever method of encoding the
sign of the wavelet coefficients with the significance information. There are
also further details of the priority of wavelet coefficients, the bit-slice coding, and adaptive arithmetic coding of quantized values (entropy coding),
which we will not pursue further in this review. The interested reader is
referred to [71] for more details.
Said and Pearlman [72] have produced an enhanced implementation of
the zerotree algorithm, known as Set Partitioning in Hierarchical Trees
(SPHIT). Their method is based on the same premises as the Shapiro algorithm, but with more attention to detail. The public domain version of this
coder is very fast, and improves the performance of EZW by 0.3-0.6 dB.
This gain is mostly due to the fact that the original zerotree algorithms
allow special symbols only for single zerotrees, while in reality, there are
other sets of zeros that appear with sufficient frequency to warrant special symbols of their own. In particular, the Said-Pearlman coder provides
symbols for combinations of parallel zerotrees.
Davis and Chawla [73] have shown that both the Shapiro and the Said
and Pearlman coders are members of a large family of tree-structured significance mapping schemes. They provide a theoretical framework that explains in more detail the performance of these coders and describe an algorithm for selecting a member of this family of significance maps that is
optimized for a given image or class of images.
46
Geoffrey M. Davis, Aria Nosratinia
7.2
Zerotrees and Rate-Distortion Optimization
In the previous coders, zerotrees were used only when they were detected
in the actual data. But consider for the moment the following hypothetical
example: assume that in an image, there is a wide area of little activity, so
that in the corresponding location of the wavelet coefficients there exists a
large group of insignificant values. Ordinarily, this would warrant the use
of a big zerotree and a low expenditure of bitrate over that area. Suppose,
however, that there is a one-pixel discontinuity in the middle of the area,
such that at the bottom of the would-be zerotree, there is one significant
coefficient. The algorithms described so far would prohibit the use of a
zerotree for the entire area.
Inaccurate representation of a single pixel will change the average distortion in the image only by a small amount. In our example we can gain
significant coding efficiency by ignoring the single significant pixel so that
we can use a large zerotree. We need a way to determine the circumstances
under which we should ignore significant coefficients in this manner.
The specification of a zerotree for a group of wavelet coefficient is a form
of quantization. Generally, the values of the pixels we code with zerotrees
are non-zero, but in using a zerotree we specify that they be decoded as zeros. Non-zerotree wavelet coefficients (significant values) are also quantized,
using scalar quantizers. If we saves bitrate by specifying larger zerotrees,
as in the hypothetical example above, the rate that was saved can be assigned to the scalar quantizers of the remaining coefficients, thus quantizing
them more accurately. Therefore, we have a choice in allocating the bitrate
among two types of quantization. The question is, if we are given a unit of
rate to use in coding, where should it be invested so that the corresponding
reduction in distortion is maximized?
This question, in the context of zerotree wavelet coding, was addressed
by Xiong et al. [74], using well-known bit allocation techniques [1]. The central result for optimal bit allocation states that, in the optimal state, the
slope of the operational rate-distortion curves of all quantizers are equal.
This result is intuitive and easy to understand. The slope of the operational rate-distortion function for each quantizer tells us how many units
of distortion we add/eliminate for each unit of rate we eliminate/add. If
one of the quantizers has a smaller R-D slope, meaning that it is giving
us less distortion reduction for our bits spent, we can take bits away from
this quantizer (i.e. we can reduce its step size) and give them to the other,
more efficient quantizers. We continue to do so until all quantizers have an
equal slope.
Obviously, specification of zerotrees affects the quantization levels of nonzero coefficients because total available rate is limited. Conversely, specifying quantization levels will affect the choice of zerotrees because it affects
the incremental distortion between zerotree quantization and scalar quantization. Therefore, an iterative algorithm is needed for rate-distortion opti-
1. Wavelet-based Image Coding: An Overview
47
mization. In phase one, the uniform scalar quantizers are fixed, and optimal
zerotrees are chosen. In phase two, zerotrees are fixed and the quantization
level of uniform scalar quantizers is optimized. This algorithm is guaranteed
to converge to a local optimum [74].
There are further details of this algorithm involving prediction and description of zerotrees, which we leave out of the current discussion. The
advantage of this method is mainly in performance, compared to both
EZW and SPHIT (the latter only slightly). The main disadvantages of this
method are its complexity, and perhaps more importantly, that it does not
generate an embedded bitstream.
8 Frequency, space-frequency adaptive coders
8.1
Wavelet Packets
The wavelet transform does a good job of decorrelating image pixels in practice, especially when images have power spectra that decay approximately
uniformly and exponentially. However, for images with non-exponential
rates of spectral decay and for images which have concentrated peaks in
the spectra away from DC, we can do considerably better.
Our analysis of Section 3.5 suggests that the optimal subband decomposition for an image is one for which the spectrum in each subband is approximately flat. The octave-band decomposition produced by the wavelet
transform produces nearly flat spectra for exponentially decaying spectra.
The Barbara test image shown in Figure 16 contains a narrow-band component at high frequencies that comes from the tablecloth and the striped
clothing. Fingerprint images contain similar narrow-band high frequency
components.
The best basis algorithm, developed by Coifman and Wickerhauser [75],
provides an efficient way to find a fast, wavelet-like transform that provides
a good approximation to the Karhunen-Loève transform for a given image.
As with the wavelet transform, we start by assuming that a given signal
corresponds to a sum of fine-scale scaling functions. The transform performs
a change of basis, but the new basis functions are not wavelets but rather
wavelet packets [76].
Like wavelets, wavelet packets are formed from translated and dilated
linear combinations of scaling functions. However, the recurrence relations
they satisfy are different, and the functions form an overcomplete set. Consider a signal of length 2N . The wavelet basis for such a signal consists of
a scaling function and 2N − 1 translates and dilates of the wavelet ψ(x).
Wavelet packets are formed from translates and dilates of 2N different prototype functions, and there are N 2N different possible functions that can
be used to form a basis.
Wavelet packets are formed from recurrence relations similar to those for
48
Geoffrey M. Davis, Aria Nosratinia
wavelets and generalize the theoretical framework of wavelets. The simplest
wavelet packet π0 (x) is the scaling function φ(x). New wavelet packets πj (x)
for j > 0 are formed by the recurrence relations
π2j (x) =
hk πj (2x − k)
(1.31)
k
π2j+1 (x) =
gk πj (2x − k).
(1.32)
k
where the hk and gk are the same as those in the recurrence equations
( 4.3) and ( 4.3).
The idea of wavelet packets is most easily seen in the frequency domain. Recall from Figure 14 that each step of the wavelet transform splits
the current low frequency subband into two subbands of equal width, one
high-pass and one low-pass. With wavelet packets there is a new degree of
freedom in the transform. Again there are N stages to the transform for a
signal of length 2N , but at each stage we have the option of splitting the
low-pass subband, the high-pass subband, both, or neither. The high and
low pass filters used in each case are the same filters used in the wavelet
transform. In fact, the wavelet transform is the special case of a wavelet
packet transform in which we always split the low-pass subband. With this
increased flexibility we can generate 2N possible different transforms in 1D. The possible transforms give rise to all possible dyadic partitions of the
frequency axis. The increased flexibility does not lead to a large increase
in complexity; the worst-case complexity for a wavelet packet transform is
O(N log N ).
8.2
Frequency Adaptive Coders
The best basis algorithm is a fast algorithm for minimizing an additive cost
function over the set of all wavelet packet bases. Our analysis of transform
coding for Gaussian random processes suggests that we select the basis that
maximizes the transform coding gain. The approximation theoretic arguments of Mallat and Falzon [57] suggest that at low bit rates the basis that
maximizes the number of coefficients below a given threshold is the best
choice. The best basis paradigm can accommodate both of these choices.
See [77] for an excellent introduction to wavelet packets and the best basis
algorithm. Ramchandran and Vetterli [65] describe an algorithm for finding
the best wavelet packet basis for coding a given image using rate-distortion
criteria.
An important application of this wavelet-packet transform optimization
is the FBI Wavelet/Scalar Quantization Standard for fingerprint compression. The standard uses a wavelet packet decomposition for the transform
stage of the encoder [78]. The transform used is fixed for all fingerprints,
however, so the FBI coder is a first-generation linear coder.
1. Wavelet-based Image Coding: An Overview
49
The benefits of customizing the transform on a per-image basis depend
considerably on the image. For the Lena test image the improvement in
peak signal to noise ratio is modest, ranging from 0.1 dB at 1 bit per pixel
to 0.25 dB at 0.25 bits per pixel. This is because the octave band partitions
of the spectrum of the Lena image are nearly flat. The Barbara image (see
Figure 16), on the other hand, has a narrow-band peak in the spectrum at
high frequencies. Consequently, the PSNR increases by roughly 2 dB over
the same range of bitrates [65]. Further impressive gains are obtained by
combining the adaptive transform with a zerotree structure [67].
8.3
Space-Frequency Adaptive Coders
The best basis algorithm is not limited only to adaptive segmentation of the
frequency domain. Related algorithms permit joint time and frequency segmentations. The simplest of these algorithms performs adapted frequency
segmentations over regions of the image selected through a quadtree decomposition procedure [79, 80]. More complicated algorithms provide combinations of spatially varying frequency decompositions and frequency varying
spatial decompositions [66]. These jointly adaptive algorithms work particularly well for highly nonstationary images.
The primary disadvantage of these spatially adaptive schemes are that
the pre-computation requirements are much greater than for the frequency
adaptive coders, and the search is also much larger. A second disadvantage is that both spatial and frequency adaptivity are limited to dyadic
partitions. A limitation of this sort is necessary for keeping the complexity
manageable, but dyadic partitions are not in general the best ones.
9 Utilizing Intra-band Dependencies
The development of the EZW coder motivated a flurry of activity in the
area of zerotree wavelet algorithms. The inherent simplicity of the zerotree
data structure, its computational advantages, as well as the potential for
generating an embedded bitstream were all very attractive to the coding
community. Zerotree algorithms were developed for a variety of applications, and many modifications and enhancements to the algorithm were
devised, as described in Section 7.
With all the excitement incited by the discovery of EZW, it is easy
to automatically assume that zerotree structures, or more generally interband dependencies, should be the focal point of efficient subband image
compression algorithms. However, some of the best performing subband
image coders known today are not based on zerotrees. In this section, we
explore two methods that utilize intra-band dependencies. One of them
uses the concept of Trellis Coded Quantization (TCQ). The other uses both
50
Geoffrey M. Davis, Aria Nosratinia
D0
D0
D1
D2
D3
D0
D1
D2
D3
D2
D0
D2
S
0
D1
D3
D1
D3
S
1
FIGURE 21. TCQ sets and supersets
inter- and intra-band information, and is based on a recursive estimation of
the variance of the wavelet coefficients. Both of them yield excellent coding
results.
9.1
Trellis coded quantization
Trellis Coded Quantization (TCQ) [81] is a fast and effective method of
quantizing random variables. Trellis coding exploits correlations between
variables. More interestingly, it can use non-rectangular quantizer cells that
give it quantization efficiencies not attainable by scalar quantizers. The
central ideas of TCQ grew out of the ground-breaking work of Ungerboeck
[82] in trellis coded modulation. In this section we describe the operational
principles of TCQ, mostly through examples. We will briefly touch upon
variations and improvements on the original idea, especially at the low
bitrates applicable in image coding. In Section 9.2, we review the use of
TCQ in multiresolution image compression algorithms.
The basic idea behind TCQ is the following: Assume that we want to
quantize a stationary, memoryless uniform source at the rate of R bits
per sample. Performing quantization directly on this uniform source would
require an optimum scalar quantizer with 2N reproduction levels (symbols).
The idea behind TCQ is to first quantize the source more finely, with 2R+k
symbols. Of course this would exceed the allocated rate, so we cannot have
a free choice of symbols at all times.
In our example we take k = 1. The scalar codebook of 2R+1 symbols is
partitioned into subsets of 2R−1 symbols each, generating four sets. In our
example R = 2; see Figure 21. The subsets are designed such that each of
them represents reproduction points of a coarser, rate-(R − 1) quantizer.
Thefour subsets are designated
D0 , D1 , D2 , and D3 . Also, define S0 =
D0 D2 and S1 = D1 D3 , where S0 and S1 are known as supersets.
Obviously, the rate constraint prohibits the specification of an arbitrary
symbol out of 2R+1 symbols. However, it is possible to exactly specify, with
R bits, one element out of either S0 or S1 . At each sample, assuming we
know which one of the supersets to use, one bit can be used to determine
the active subset, and R − 1 bits to specify a codeword from the subset.
The choice of superset is determined by the state of a finite state machine,
described by a suitable trellis. An example of such a trellis, with eight
1. Wavelet-based Image Coding: An Overview
51
D0 D2
D1 D3
D0 D2
D1 D3
D2 D0
D3 D1
D2 D0
D3 D1
FIGURE 22. An 8-state TCQ trellis with subset labeling. The bits that specify the
sets within the superset also dictate the path through the trellis.
states, is given in Figure 22. The subsets {D0 , D1 , D2 , D3 } are also used to
label the branches of the trellis, so the same bit that specifies the subset
(at a given state) also determines the next state of the trellis.
Encoding is achieved by spending one bit per sample on specifying the
path through the trellis, while the remaining R − 1 bits specify a codeword
out of the active subset. It may seem that we are back to a non-optimal
rate-R quantizer (either S0 or S1 ). So why all this effort? The answer
is that we have more codewords than a rate-R quantizer, because there
is some freedom of choosing from symbols of either S0 or S1 . Of course
this choice is not completely free: the decision made at each sample is
linked to decisions made at past and future sample points, through the
permissible paths of the trellis. But it is this additional flexibility that
leads to the improved performance. Availability of both S0 and S1 means
that the reproduction levels of the quantizer are, in effect, allowed to “slide
around” and fit themselves to the data, subject to the permissible paths
on the trellis.
Before we continue with further developments of TCQ and subband coding, we should note that in terms of both efficiency and computational
requirements, TCQ is much more similar to VQ than to scalar quantization. Since our entire discussion of transform coding has been motivated by
an attempt to avoid VQ, what is the motivation for using TCQ in subband
coding, instead of standard VQ? The answer lies in the recursive structure of trellis coding and the existence of a simple dynamic programming
method, known as the Viterbi algorithm [83], for finding the TCQ codewords. Although it is true that block quantizers, such as VQ, are asymptotically as efficient as TCQ, the process of approaching the limit is far from
trivial for VQ. For a given realization of a random process, the code vectors
generated by the VQ of size N − 1 have no clear relationship to those with
vector dimension N . In contrast, the trellis encoding algorithm increases
52
Geoffrey M. Davis, Aria Nosratinia
the dimensionality of the problem automatically by increasing the length
of the trellis.
The standard version of TCQ is not particularly suitable for image coding, because its performance degrades quickly at low rates. This is due
partially to the fact that one bit per sample is used to encode the trellis
alone, while interesting rates for image coding are mostly below one bit per
sample. Entropy constrained TCQ (ECTCQ) improves the performance of
TCQ at low rates. In particular, a version of ECTCQ due to Marcellin [84]
addresses two key issues: reducing the rate used to represent the trellis
(the so-called “state entropy”), and ensuring that zero can be used as an
output codeword with high probability. The codebooks are designed using
the algorithm and encoding rule from [85].
9.2
TCQ subband coders
In a remarkable coincidence, at the 1994 International Conference in Image
Processing, three research groups [86, 87, 88] presented similar but independently developed image coding algorithms. The main ingredients of the
three methods are subband decomposition, classification and optimal rate
allocation to different subsets of subband data, and entropy-constrained
TCQ. These works have been brought together in [68]. We briefly discuss
the main aspects of these algorithms.
Consider a subband decomposition of an image, and assume that the subbands are well represented by a non-stationary random process X, whose
samples Xi are taken from distributions with variances σi2 . One can compute an “average variance” over the entire random process and perform
conventional optimal quantization. But better performance is possible by
sending overhead information about the variance of each sample, and quantizing it optimally according to its own p.d.f.
This basic idea was first proposed by Chen and Smith [89] for adaptive
quantization of DCT coefficients. In their paper, Chen and Smith proposed
to divide all DCT coefficients into four groups according to their “activity level”, i.e. variance, and code each coefficient with an optimal quantizer designed for its group. The question of how to partition coefficients
into groups was not addressed, however, and [89] arbitrarily chose to form
groups with equal population.9
However, one can show that equally populated groups are not a always
9 If for a moment, we disregard the overhead information, the problem of partitioning
the coefficients bears a strong resemblance to the problem of best linear transform.
Both operations, namely the linear transform and partitioning, conserve energy. The
goal in both is to minimize overall distortion through optimal allocation of a finite rate.
Not surprisingly, the solution techniques are similar (Lagrange multipliers), and they
both generate sets with maximum separation between low and high energies (maximum
arithmetic to geometric mean ratio).
1. Wavelet-based Image Coding: An Overview
53
a good choice. Suppose that we want to classify the samples into J groups,
and that all samples assigned to a given class i ∈ {1, ..., J} are grouped into
a source Xi . Let the total number of samples assigned to Xi be Ni , and the
total number of samples in all groups be N . Define pi = Ni /N to be the
probability of a sample belonging to the source Xi . Encoding the source
Xi at rate Ri results in a mean squared error distortion of the form [90]
Di (Ri ) = 52i σi2 2−2Ri
(1.33)
where 5i is a constant depending on the shape of the pdf. The rate allocation
problem can now be solved using a Lagrange multiplier approach, much in
the same way as was shown for optimal linear transforms, resulting in the
following optimal rates:
Ri =
52 σ 2
R 1
+ log2 J i i
2 2 pj
J
2
j=1 (5j σj )
(1.34)
where R is the total rate and Ri are the rates assigned to each group.
Classification gain is defined as the ratio of the quantization error of the
original signal X, divided by that of the optimally bit-allocated classified
version.
52 σ 2
Gc = J
(1.35)
2 2 pj
j=1 (5j σj )
One aims to maximize this gain over {pi }. It is not unexpected that
the optimization process can often yield non-uniform {pi }, resulting in
unequal population of the classification groups. It is noteworthy that nonuniform populations not only have better classification gain in general, but
also lower overhead: Compared to a uniform {pi }, any other distribution
has smaller entropy, which implies smaller side information to specify the
classes.
The classification gain is defined for Xi taken from one subband. A generalization of this result in [68] combines it with the conventional coding
gain of the subbands. Another refinement takes into account the side information required for classification. The coding algorithm then optimizes
the resulting expression to determine the classifications. ECTCQ is then
used for final coding.
Practical implementation of this algorithm requires attention to a great
many details, for which the interested reader is referred to [68]. For example,
the classification maps determine energy levels of the signal, which are
related to the location of the edges in the image, and are thus related in
different subbands. A variety of methods can be used to reduce the overhead
information (in fact, the coder to be discussed in the next section makes the
management of side information the focus of its efforts) Other issues include
alternative measures for classification, and the usage of arithmetic coded
TCQ. The coding results of the ECTCQ based subband coding are some of
54
Geoffrey M. Davis, Aria Nosratinia
the best currently available in the literature, although the computational
complexity of these algorithms is also considerably greater than the other
methods presented in this paper.
9.3
Mixture Modeling and Estimation
A common thread in successful subband and wavelet image coders is modeling of image subbands as random variables drawn from a mixture of
distributions. For each sample, one needs to detect which p.d.f. of the mixture it is drawn from, and then quantize it according to that pdf. Since
the decoder needs to know which element of the mixture was used for encoding, many algorithms send side information to the decoder. This side
information becomes significant, especially at low bitrates, so that efficient
management of it is pivotal to the success of the image coder.
All subband and wavelet coding algorithms discussed so far use this idea
in one way or another. They only differ in the constraints they put on side
information so that it can be coded efficiently. For example, zerotrees are
a clever way of indicating side information. The data is assumed from a
mixture of very low energy (zero set) and high energy random variables,
and the zero sets are assumed to have a tree structure.
The TCQ subband coders discussed in the last section also use the same
idea. Different classes represent different energies in the subbands, and are
transmitted as overhead. In [68], several methods are discussed to compress the side information, again based on geometrical constraints on the
constituent elements of the mixture (energy classes).
A completely different approach to the problem of handling information
overhead is explored in [69, 91]. These two works were developed simultaneously but independently. The version developed in [69] is named Estimation
Quantization (EQ) by the authors, and is the one that we present in the
following. The title of [91] suggests a focus on entropy coding, but in fact
the underlying ideas of the two are remarkably similar. We will refer to the
the aggregate class as backward mixture-estimation encoding (BMEE).
BMEE models the wavelet subband coefficients as non-stationary generalized Gaussian, whose non-stationarity is manifested by a slowly varying
variance (energy) in each band. Because the energy varies slowly, it can be
predicted from causal neighboring coefficients. Therefore, unlike previous
methods, BMEE does not send the bulk of mixture information as overhead, but attempts to recover it at the decoder from already transmitted
data, hence the designation “backward”. BMEE assumes that the causal
neighborhood of a subband coefficient (including parents in a subband tree)
has the same energy (variance) as the coefficient itself. The estimate of energy is found by applying a maximum likelihood method to a training set
formed by the causal neighborhood.
Similar to other recursive algorithms that involve quantization, BMEE
has to contend with the problem of stability and drift. Specifically, the
1. Wavelet-based Image Coding: An Overview
55
decoder has access only to quantized coefficients, therefore the estimator of
energy at the encoder can only use quantized coefficients. Otherwise, the
estimates at the encoder and decoder will vary, resulting in drift problems.
This presents the added difficulty of estimating variances from quantized
causal coefficients. BMEE incorporates the quantization of the coefficients
into the maximum likelihood estimation of the variance.
The quantization itself is performed with a dead-zone uniform quantizer
(see Figure 15). This quantizer offers a good approximation to entropy constrained quantization of generalized Gaussian signals. The dead-zone and
step sizes of the quantizers are determined through a Lagrange multiplier
optimization technique, which was introduced in the section on optimal
rate allocation. This optimization is performed offline, once each for a variety of encoding rates and shape parameters, and the results are stored in
a look-up table. This approach is to be credited for the speed of the algorithm, because no optimization need take place at the time of encoding the
image.
Finally, the backward nature of the algorithm, combined with quantization, presents another challenge. All the elements in the causal neighborhood may sometimes quantize to zero. In that case, the current coefficient
will also quantize to zero. This degenerate condition will propagate through
the subband, making all coefficients on the causal side of this degeneracy
equal to zero. To avoid this condition, BMEE provides for a mechanism to
send side information to the receiver, whenever all neighboring elements
are zero. This is accomplished by a preliminary pass through the coefficients, where the algorithm tries to “guess” which one of the coefficients
will have degenerate neighborhoods, and assembles them to a set. From
this set, a generalized Gaussian variance and shape parameter is computed
and transmitted to the decoder. Every time a degenerate case happens, the
encoder and decoder act based on this extra set of parameters, instead of
using the backward estimation mode.
The BMEE coder is very fast, and especially in the low bitrate mode
(less than 0.25 bits per pixel) is extremely competitive. This is likely to
motivate a re-visitation of the role of side information and the mechanism
of its transmission in wavelet coders.
10 Future Trends
Current research in image coding is progressing along a number of fronts.
At the most basic level, a new interpretation of the wavelet transform
has appeared in the literature. This new theoretical framework, called the
lifting scheme [41], provides a simpler and more flexible method for designing wavelets than standard Fourier-based methods. New families of nonseparable wavelets constructed using lifting have the potential to improve
56
Geoffrey M. Davis, Aria Nosratinia
coders. One very intriguing avenue for future research is the exploration of
the nonlinear analogs of the wavelet transform that lifting makes possible.
The area of classification and backward estimation based coders is an
active one. Several research groups are reporting promising results [92, 93].
One very promising research direction is the development of coded images
that are robust to channel noise via joint source and channel coding. See
for example [94], [95] and [96].
The adoption of wavelet based coding to video signals presents special
challenges. One can apply 2-D wavelet coding in combination to temporal
prediction (motion estimated prediction), which will be a direct counterpart
of current DCT-based video coding methods. It is also possible to consider
the video signal as a three-dimensional array of data and attempt to compress it with 3-D wavelet analysis. This approach presents difficulties that
arise from the fundamental properties of the discrete wavelet transform.
The discrete wavelet transform (as well as any subband decomposition) is
a space-varying operator, due to the presence of decimation and interpolation. This space variance is not conducive to compact representation of
video signals, as described below.
Video signals are best modeled by 2-D projections whose position in
consecutive frames of the video signal varies by unknown amounts. Because
vast amounts of information are repeated in this way, one can achieve
considerable gain by representing the repeated information only once. This
is the basis of motion compensated coding. However, since the wavelet
representation of the same 2-D signal will vary once it is shifted10 , this
redundancy is difficult to reproduce in the wavelet domain. A frequency
domain study of the difficulties of 3-D wavelet coding of video is presented
in [97], and leads to the same insights. Some attempts have also been made
on applying 3-D wavelet coding on the residual 3-D data after motion
compensation, but have met with indifferent success.
11 Summary and Conclusion
Image compression is governed by the general laws of information theory
and specifically rate-distortion theory. However, these general laws are nonconstructive and the more specific techniques of quantization theory are
needed for the actual development of compression algorithms.
Vector quantization can theoretically attain the maximum achievable
coding efficiency. However, VQ has three main impediments: computational
complexity, delay, and the curse of dimensionality. Transform coding techniques, in conjunction with entropy coding, capture important gains of VQ,
10 Unless the shift is exactly by a correct multiple of M samples, where M is the
downsampling rate
1. Wavelet-based Image Coding: An Overview
57
while avoiding most of its difficulties.
Theoretically, the Karhunen-Loéve transform is optimal for Gaussian
processes. Approximations to the K-L transform, such as the DCT, have
led to very successful image coding algorithms such as JPEG. However,
even if one argues that image pixels can be individually Gaussian, they
cannot be assumed to be jointly Gaussian, at least not across the image
discontinuities. Image discontinuities are the place where traditional coders
spend the most rate, and suffer the most distortion. This happens because
traditional Fourier-type transforms (e.g., DCT) disperse the energy of discontinuous signals across many coefficients, while the compaction of energy
in the transform domain is essential for good coding performance.
The discrete wavelet transform provides an elegant framework for signal
representation in which both smooth areas and discontinuities can be represented compactly in the transform domain. This ability comes from the
multi-resolution properties of wavelets. One can motivate wavelets through
spectral partitioning arguments used in deriving optimal quantizers for
Gaussian processes. However, the usefulness of wavelets in compression
goes beyond the Gaussian case.
State of the art wavelet coders assume that image data comes from a
source with fluctuating variance. Each of these coders provides a mechanism
to express the local variance of the wavelet coefficients, and quantizes the
coefficients optimally or near-optimally according to that variance. The
individual wavelet coders vary in the way they estimate and transmit this
variances to the decoder, as well as the strategies for quantizing according
to that variance.
Zerotree coders assume a two-state structure for the variances: either
negligible (zero) or otherwise. They send side information to the decoder
to indicate the positions of the non-zero coefficients. This process yields
a non-linear image approximation rather than the linear truncated KLTbased approximation motivated by our Gaussian model. The set of zero coefficients are expressed in terms of wavelet trees (Lewis & Knowles, Shapiro,
others) or combinations thereof (Said & Pearlman). The zero sets are transmitted to the receiver as overhead, as well as the rest of the quantized data.
Zerotree coders rely strongly on the dependency of data across scales of the
wavelet transform.
Frequency-adaptive coders improve upon basic wavelet coders by adapting transforms according to the local inter-pixel correlation structure within
an image. Local fluctuations in the correlation structure and in the variance
can be addressed by spatially adapting the transform and by augmenting
the optimized transforms with a zerotree structure.
Other wavelet coders use dependency of data within the bands (and
sometimes across the bands as well). Coders based on Trellis Coded Quantization (TCQ) partition coefficients into a number of groups, according to
their energy. For each coefficient, they estimate and/or transmit the group
information as well as coding the value of the coefficient with TCQ, ac-
58
Geoffrey M. Davis, Aria Nosratinia
cording to the nominal variance of the group. Another newly developed
class of coders transmit only minimal variance information while achieving
impressive coding results, indicating that perhaps the variance information
is more redundant than previously thought.
While some of these coders may not employ what might strictly be called
a wavelet transform, they all utilize a multi-resolution decomposition, and
use concepts that were motivated by wavelet theory. Wavelets and the ideas
arising from wavelet analysis have had an indelible effect on the theory and
practice of image compression, and are likely to continue their dominant
presence in image coding research in the near future.
Acknowledgments: G. Davis thanks the Digital Signal Processing group at
Rice University for their generous hospitality during the writing of this
paper. This work has been supported in part by a Texas Instruments Visiting Assistant Professorship at Rice University and an NSF Mathematical
Sciences Postdoctoral Research Fellowship.
12
References
[1] A. Gersho and R. M. Gray, Vector Quantization and Signal Compression. Kluwer Academic Publishers, 1992.
[2] C. E. Shannon, “Coding theorems for a discrete source with a fidelity
criterion,” in IRE Nat. Conv. Rec., vol. 4, pp. 142–163, March 1959.
[3] R. M. Gray, J. C. Kieffer, and Y. Linde, “Locally optimal block quantizer design,” Information and Control, vol. 45, pp. 178–198, May 1980.
[4] K. Zeger, J. Vaisey, and A. Gersho, “Globally optimal vector quantizer design by stochastic relaxation,” IEEE Transactions on Signal
Processing, vol. 40, pp. 310–322, Feb. 1992.
[5] T. Cover and J. Thomas, Elements of Information Theory. New York:
John Wiley & Sons, Inc., 1991.
[6] A. Gersho, “Asymptotically optimal block quantization,” IEEE Transactions on Information Theory, vol. IT-25, pp. 373–380, July 1979.
[7] L. F. Toth, “Sur la representation d’une population infinie par un nombre fini d’elements,” Acta Math. Acad. Scient. Hung., vol. 10, pp. 299–
304, 1959.
[8] P. L. Zador, “Asymptotic quantization error of continuous signals and
the quantization dimension,” IEEE Transactions on Information Theory, vol. IT-28, pp. 139–149, Mar. 1982.
1. Wavelet-based Image Coding: An Overview
59
[9] J. Y. Huang and P. M. Schultheiss, “Block quantization of correlated
Gaussian random variables,” IEEE Transactions on Communications,
vol. CS-11, pp. 289–296, September 1963.
[10] R. A. Horn and C. R. Johnson, Matrix Analysis. New York: Cambridge
University Press, 1990.
[11] N. Ahmed, T. Natarajan, and K. R. Rao, “Descrete cosine transform,”
IEEE Transactions on Computers, vol. C-23, pp. 90–93, January 1974.
[12] K. Rao and P. Yip, The Discrete Cosine Transform. New York: Academic Press, 1990.
[13] W. Pennebaker and J. Mitchell, JPEG still image data compression
standard. New York: Van Nostrad Reinhold, 1993.
[14] Y. Arai, T. Agui, and M. Nakajima, “A fast DCT-SQ scheme for
images,” Transactions of the IEICE, vol. 71, p. 1095, November 1988.
[15] E. Feig and E. Linzer, “Discrete cosine transform algorithms for image data compression,” in Electronic Imaging ’90 East, (Boston, MA),
pp. 84–87, October 1990.
[16] A. Croisier, D. Esteban, and C. Galand, “Perfect channel splitting
by use of interpolation/decimation/tree decomposition techniques,”
in Proc. Int. Symp. on Information, Circuits and Systems, (Patras,
Greece), 1976.
[17] P. Vaidyanathan, “Theory and design of M-channel maximally decimated quadrature mirror filters with arbitrary M, having perfect reconstruction property,” IEEE Trans. ASSP, vol. ASSP-35, pp. 476–
492, April 1987.
[18] M. J. T. Smith and T. P. Barnwell, “A procedure for desiging exact
reconstruction fiolter banks for tree structured subband coders,” in
Proc. ICASSP, (San Diego, CA), pp. 27.1.1–27.1.4, March 1984.
[19] M. J. T. Smith and T. P. Barnwell, “Exact reconstruction techniques
for tree-structured subband coders,” IEEE Trans. ASSP, vol. ASSP34, pp. 434–441, June 1986.
[20] M. Vetterli, “Splitting a signal into subband channels allowing perfect reconstruction,” in Proc. IASTED Conf. Appl. Signal Processing,
(Paris, France), June 1985.
[21] M. Vetterli, “Filter banks allowing perfect reconstruction,” Signal Processing, vol. 10, pp. 219–244, April 1986.
[22] P. Vaidyanathan, Multirate systems and filter banks. Englewood Cliffs,
NJ: Prentice Hall, 1993.
60
Geoffrey M. Davis, Aria Nosratinia
[23] M. Vetterli and J. Kovačević, Wavelets and Subband Coding. Englewood Cliffs, NJ: Prentice Hall, 1995.
[24] T. Berger, Rate Distortion Theory. Englewood Cliffs, NJ: Prentice
Hall, 1971.
[25] B. Girod, F. Hartung, and U. Horn, “Subband image coding,” in Subband and wavelet transforms: design and applications, Boston, MA:
Kluwer Academic Publishers, 1995.
[26] P. J. Burt and E. H. Adelson, “The Laplacian pyramid as a compact
image code,” IEEE Transactions on Communications, vol. COM-31,
Apr. 1983.
[27] S. Mallat, “A theory for multiresolution signal decomposition: the
wavelet representation,” IEEE Trans. Pattern Analysis and Machine
Intelligence, vol. 11, pp. 674–693, July 1989.
[28] G. Strang and T. Q. Nguyen, Wavelets and Filter Banks. Wellesley,
MA: Wellesley-Cambridge Press, 1996.
[29] C. S. Burrus, R. A. Gopinath, and H. Guo, Introduction to Wavelets
and Wavelet Transforms: A Primer. Englewood Cliffs, NJ: PrenticeHall, 1997.
[30] I. Daubechies, Ten Lectures on Wavelets. Philadelphia, PA: SIAM,
1992.
[31] O. Rioul, “Simple regularity criteria for subdivision schemes,” SIAM
J. Math. Analysis, vol. 23, pp. 1544–1576, Nov. 1992.
[32] O. Rioul, “Regular wavelets: a discrete-time approach,” IEEE Transactions on Signal Processing, vol. 41, pp. 3572–3579, Dec. 1993.
[33] G. Strang and G. Fix, “A Fourier analysis of the finite element variational method,” Constructive Aspects of Functional Analysis, pp. 796–
830, 1971.
[34] W. Sweldens and R. Piessens, “Quadrature formulae and asymptotic
error expansions for wavelet approximations of smooth functions,”
SIAM Journal of Numerical Analysis, vol. 31, pp. 1240–1264, Aug.
1994.
[35] M. Unser, “Approximation power of biorthogonal wavelet expansions,”
IEEE Transactions on Signal Processing, vol. 44, pp. 519–527, Mar.
1996.
[36] J. Kovačević and W. Sweldens, “Interpolating filter banks and wavelets
in arbitrary dimensions,” tech. rep., Lucent Technologies, Murray Hill,
NJ, 1997.
1. Wavelet-based Image Coding: An Overview
61
[37] M. Antonini, M. Barlaud, and P. Mathieu, “Image Coding Using
Wavelet Transform,” IEEE Trans. Image Proc., vol. 1, pp. 205–220,
Apr. 1992.
[38] M. Vetterli and C. Herley, “Wavelets and filter banks: Theory and
design,” IEEE Trans. Acoust. Speech Signal Proc., vol. 40, no. 9,
pp. 2207–2232, 1992.
[39] J. Villasenor, B. Belzer, and J. Liao, “Wavelet filter evaluation for
image compression,” IEEE Transactions on image processing, vol. 2,
pp. 1053–1060, Aug. 1995.
[40] G. Deslauriers and S. Dubuc, “Symmetric iterative interpolation processes,” Constructive Approximation, vol. 5, no. 1, pp. 49–68, 1989.
[41] W. Sweldens, “The lifting scheme: A new philosophy in biorthogonal
wavelet constructions,” in Wavelet Applications in Signal and Image
Processing III (A. F. Laine and M. Unser, eds.), pp. 68–79, Proc.
SPIE 2569, 1995.
[42] I. Balasingham and T. A. Ramstad, “On the relevance of the regularity
constraint in subband image coding,” in Proc. Asilomar Conference on
Signals, Systems, and Computers, (Pacific Grove), 1997.
[43] C. M. Brislawn, “Classification of nonexpansive symmetric extension
transforms for multirate filter banks,” Applied and Comp. Harmonic
Analysis, vol. 3, pp. 337–357, 1996.
[44] C. Herley and M. Vetterli, “Orthogonal time-varying filter banks and
wavelets,” in Proc. IEEE Int. Symp. Circuits Systems, vol. 1, pp. 391–
394, May 1993.
[45] C. Herley, “Boundary filters for finite-length signals and time-varying
filter banks,” IEEE Trans. Circuits and Systems II, vol. 42, pp. 102–
114, Feb. 1995.
[46] W. Sweldens and P. Schröder, “Building your own wavelets at home,”
Tech. Rep. 1995:5, Industrial Mathematics Initiative, Mathematics Department, University of South Carolina, 1995.
[47] H. Gish and J. N. Pierce, “Asymptotically efficient quantizing,” IEEE
Transactions on Information Theory, vol. IT-14, pp. 676–683, Sept.
1968.
[48] N. Farvardin and J. W. Modestino, “Optimum quantizer performance
for a class of non-Gaussian memoryless sources,” IEEE Transactions
on Information Theory, vol. 30, pp. 485–497, May 1984.
62
Geoffrey M. Davis, Aria Nosratinia
[49] D. Taubman and A. Zakhor, “Multirate 3-D subband coding of video,”
IEEE Trans. Image Proc., vol. 3, Sept. 1994.
[50] T. Bell, J. G. Cleary, and I. H. Witten, Text Compression. Englewood
Cliffs, NJ: Prentice Hall, 1990.
[51] D. L. Duttweiler and C. Chamzas, “Probability estimation in arithmetic and adaptive-Huffman entropy coders,” IEEE Transactions on
Image Processing, vol. 4, pp. 237–246, Mar. 1995.
[52] Y. Shoham and A. Gersho, “Efficient bit allocation for an arbitrary set
of quantizers,” IEEE Trans. Acoustics, Speech, and Signal Processing,
vol. 36, pp. 1445–1453, Sept. 1988.
[53] P. Moulin, “A multiscale relaxation algorithm for SNR maximization
in nonorthogonal subband coding,” IEEE Transactions on Image Processing, vol. 4, pp. 1269–1281, Sept. 1995.
[54] J. W. Woods and T. Naveen, “A filter based allocation scheme for
subband compression of HDTV,” IEEE Trans. Image Proc., vol. IP-1,
pp. 436–440, July 1992.
[55] B. A. Wandell, Foundations of Vision. Sunderland, MA: Sinauer Associates, 1995.
[56] A. Watson, G. Yang, J. Soloman, and J. Villasenor, “Visual thresholds
for wavelet quantization error,” in Proceedings of the SPIE, vol. 2657,
pp. 382–392, 1996.
[57] S. Mallat and F. Falzon, “Understanding image transform codes,” in
Proc. SPIE Aerospace Conf., (orlando), Apr. 1997.
[58] R. A. DeVore, B. Jawerth, and B. J. Lucier, “Image Compression
through Wavelet Transform Coding,” IEEE Trans. Info. Theory,
vol. 38, pp. 719–746, Mar. 1992.
[59] W. B. Pennebaker and J. L. Mitchell, JPEG Still Image Data Compression Standard. New York: Van Nostrand Reinhold, 1992.
[60] M. Crouse and K. Ramchandran, “Joint thresholding and quantizer
selection for decoder-compatible baseline JPEG,” in Proc. ICASSP,
May 1995.
[61] G. M. Davis, “The wavelet image compression construction kit.”
http://www.cs.dartmouth.edu/∼gdavis/wavelet/wavelet.html, 1996.
[62] J. Shapiro, “Embedded image coding using zerotrees of wavelet coefficients,” IEEE Transactions on Signal Processing, vol. 41, pp. 3445–
3462, Dec. 1993.
1. Wavelet-based Image Coding: An Overview
63
[63] A. Said and W. A. Pearlman, “A new, fast, and efficient image codec
based on set partitioning in hierarchichal trees,” IEEE Trans. Circuits
and Systems for Video Technology, vol. 6, pp. 243–250, June 1996.
[64] Z. Xiong, K. Ramchandran, and M. T. Orchard, “Space-frequency
quantization for wavelet image coding,” preprint, 1995.
[65] K. Ramchandran and M. Vetterli, “Best wavelet packet bases in a
rate-distortion sense,” IEEE Transactions on Image Processing, vol. 2,
no. 2, pp. 160–175, 1992.
[66] C. Herley, Z. Xiong, K. Ramchandran, and M. T. Orchard, “Joint
space-frequency segmentation using balanced wavelet packet trees for
least-cost image representation,” IEEE Transactions on Image Processing, Sept. 1997.
[67] Z. Xiong, K. Ramchandran, M. Orchard, and K. Asai, “Wavelet
packets-based image coding using joint space-frequency quantization,”
Preprint, 1996.
[68] R. L. Joshi, H. Jafarkhani, J. H. Kasner, T. R. Fisher, N. Farvardin,
and M. W. Marcellin, “Comparison of different methods of classification in subband coding of images,” IEEE Transactions on Image
Processing, submitted.
[69] S. M. LoPresto, K. Ramchandran, and M. T. Orchard, “Image coding based on mixture modeling of wavelet coefficients and a fast
estimation-quantization framework,” in Proc. Data Compression Conference, (Snowbird, Utah), pp. 221–230, 1997.
[70] A. S. Lewis and G. Knowles, “Image compression using the 2-d wavelet
transform,” IEEE Transactions on Image Processing, vol. 1, pp. 244–
250, April 1992.
[71] J. Shapiro, “Embedded image coding using zero-trees of wavelet coefficients,” IEEE Transactions on Signal Processing, vol. 41, pp. 3445–
3462, December 1993.
[72] A. Said and W. A. Pearlman, “A new, fast, and efficient image codec
based on set partitioning in hierarchical trees,” IEEE Transactions on
Circuits and Systems for Video Technology, vol. 6, pp. 243–250, June
1996.
[73] G. M. Davis and S. Chawla, “Image coding using optimized significance tree quantization,” in Proc. Data Compression Conference (J. A.
Storer and M. Cohn, eds.), pp. 387–396, Mar. 1997.
64
Geoffrey M. Davis, Aria Nosratinia
[74] Z. Xiong, K. Ramchandran, and M. T. Orchard, “Space-frequency
quantization for wavelet image coding,” IEEE Transactions on Image
Processing, vol. 6, pp. 677–693, May 1997.
[75] R. R. Coifman and M. V. Wickerhauser, “Entropy based algorithms
for best basis selection,” IEEE Transactions on Information Theory,
vol. 32, pp. 712–718, Mar. 1992.
[76] R. R. Coifman and Y. Meyer, “Nouvelles bases orthonormées de l2 (r)
ayant la structure du système de Walsh,” Tech. Rep. Preprint, Department of Mathematics, Yale University, 1989.
[77] M. V. Wickerhauser, Adapted Wavelet Analysis from Theory to Software. Wellesley, MA: A. K. Peters, 1994.
[78] C. J. I. Services, WSQ Gray-Scale Fingerprint Image Compression
Specification (ver. 2.0). Federal Bureau of Investigation, Feb. 1993.
[79] C. Herley, J. Kovačević, K. Ramchandran, and M. Vetterli, “Tilings of
the time-frequency plane: Construction of arbitrary orthogonal bases
and fast tiling algorithms,” IEEE Transactions on Signal Processing,
vol. 41, pp. 3341–3359, Dec. 1993.
[80] J. R. Smith and S. F. Chang, “Frequency and spatially adaptive
wavelet packets,” in Proc. ICASSP, May 1995.
[81] M. W. Marcellin and T. R. Fischer, “Trellis coded quantization of
memoryless and Gauss-Markov sources,” IEEE Transactions on Communications, vol. 38, pp. 82–93, January 1990.
[82] G. Ungerboeck, “Channel coding with multilevel/phase signals,” IEEE
Transactions on Information Theory, vol. IT-28, pp. 55–67, January
1982.
[83] G. D. Forney, “The Viterbi algorithm,” Proceedings of the IEEE,
vol. 61, pp. 268–278, March 1973.
[84] M. W. Marcellin, “On entropy-constrained trellis coded quantization,”
IEEE Transactions on Communications, vol. 42, pp. 14–16, January
1994.
[85] P. A. Chou, T. Lookabaugh, and R. M. Gray, “Entropy-constrained
vector quantization,” IEEE Transactions on Information Theory,
vol. 37, pp. 31–42, January 1989.
[86] H. Jafarkhani, N. Farvardin, and C. C. Lee, “Adaptive image coding
based on the discrete wavelet transform,” in Proc. IEEE Int. Conf. Image Proc. (ICIP), vol. 3, (Austin, TX), pp. 343–347, November 1994.
1. Wavelet-based Image Coding: An Overview
65
[87] R. L. Joshi, T. Fischer, and R. H. Bamberger, “Optimum classification
in subband coding of images,” in Proc. IEEE Int. Conf. Image Proc.
(ICIP), vol. 2, (Austin, TX), pp. 883–887, November 1994.
[88] J. H. Kasner and M. W. Marcellin, “Adaptive wavelet coding of images,” in Proc. IEEE Int. Conf. Image Proc. (ICIP), vol. 3, (Austin,
TX), pp. 358–362, November 1994.
[89] W. H. Chen and C. H. Smith, “Adaptive coding of monochrome and
color images,” IEEE Transactions on Communications, vol. COM-25,
pp. 1285–1292, November 1977.
[90] N. S. Jayant and P. Noll, Digital Coding of waveforms. Englewood
Cliffs, NJ: Prentice-Hall, 1984.
[91] C. Chrysafis and A. Ortega, “Efficient context based entropy coding for lossy wavelet image compression,” in Proc. Data Compression
Conference, (Snowbird, Utah), pp. 241–250, 1997.
[92] Y. Yoo, A. Ortega, and B. Yu, “Progressive classification and adaptive
quantization of image subbands.” Preprint, 1997.
[93] D. Marpe and H. L. Cycon, “Efficient pre-coding techniques for
wavelet-based image compression.” Submitted to PCS, Berlin, 1997.
[94] P. G. Sherwood and K. Zeger, “Progressive image coding on noisy
channels,” in Proc. Data Compression Conference, (Snowbird, UT),
pp. 72–81, Mar. 1997.
[95] S. L. Regunathan, K. Rose, and S. Gadkari, “Multimode image coding
for noisy channels,” in Proc. Data Compression Conference, (Snowbird, UT), pp. 82–90, Mar. 1997.
[96] J. Garcia-Frias and J. D. Villasenor, “An analytical treatment of
channel-induced distortion in run length coded image subbands,” in
Proc. Data Compression Conference, (Snowbird, UT), pp. 52–61, Mar.
1997.
[97] A. Nosratinia and M. T. Orchard, “A multi-resolution framework for
backward motion compensation,” in Proc. SPIE Symposium on Electronic Imaging, (San Jose, CA), February 1995.
Tutorial on Hidden Markov Models
c 1995, 2002 Javier R. Movellan.
Copyright This is an open source document. Permission is granted to copy, distribute and/or modify
this document under the terms of the GNU Free Documentation License, Version 1.2 or
any later version published by the Free Software Foundation; with no Invariant Sections,
no Front-Cover Texts, and no Back-Cover Texts.
Endorsements
This document is endorsed by its copyright holder: Javier R. Movellan. Modified versions
of this document should delete this endorsement.
1 Notation for discrete HMM
S = {S1 , ..., SN }. Set of possible hidden states.
N. Number of distinct hidden states.
V = {v1 , ..., vM }. Set of possible external observations.
M. Number of distinct external observations.
o = (o1 , ..., oK ). A sample of sequences of external observations (the training sample).
Each element in o is an entire sequence of observations (e.g., a word).
K. Number of sequences (e.g., words) in the training sample.
o = (o1 , ..., oT ). A sequence of external observations (e.g., a word). If there is more than
a sequence I use a superscript l.
ot . A variable representing the external observation at time t. If there is more than a
sequence of interest, I use a superscript (e.g., o42 = 3 means that in sequence number 4, at
time 2 we observe v3 ).
T . The number of time steps in the sequence .
q = (q 1 , ..., q K ). Collection of sequences of internal states (internal state sequences that
may correspond to each sequence in the training sequence).
q = (q1 , ..., qT ). A sequence of internal states. If there is more than a sequence of interest
I use a superscript to denote the sequence of interest.
qt . A variable representing the internal state at time t. If there is more than a sequence I
use a superscript (e.g., q24 = 3 means that the system is in state S3 at time 2 in sequence
number 4 of the training sample).
λ = (A, B, π). A hidden Markov model as defined by its A, B and π matrices.
aij = Pλ (qt+1 = j|qt = i). State transition probability for model λ.
A = {aij }. The N xN matrix of transition probabilities.
bj (k) = Pλ (ot = k|qt = j). Emission probability for observation vk by state Sj .
B = {bj (k)}. The MxN matrix of state to observations probabilities.
πi = Pλ (q1 = i). Initial state probability for model λ.
π = {πi }. The vector of initial state probabilities.
P
Q(λ, λ̄) = q Pλ (q|0)log Pλ̄ (qo). The auxiliary function maximized by the E-M algorithm. λ represents the current model, λ̄ represents the new model under consideration.
Ql (λ, λ̄) The E-M function restricted to the sequence ol from the training sample.
αt (i) = Pλ (o1 ...ot qt = i). The forward variable for the sequence o at time t for state i. If
there is more than one sequence of interest I use a superscript to denote the sequence.
βt (i) = Pλ (ot+1 ...oT |qt = i). The scaled backward variable for the sequence o at time t
for state i. If there is more than one sequence of interest I use a superscript to denote the
sequence.
α̂t (i). The scaled forward variable for the sequence o at time t for state i. If there is more
than one sequence of interest I use a superscript to denote the sequence.
β̂t (i). The scaled backward variable for the sequence o at time t for state i. If there is more
than one sequence of interest I use a superscript to denote the sequence.
c1 ...cT . The scaling coefficients in the scaled forward and backward algorithm for the
sequence ol . If there is more than one sequence of interest I use a superscript to denote the
sequence.
γt (i) = Pλ (qt = i|o) If there is more than one sequence of interest I use a superscript to
denote the sequence.
ξt (i, j) = Pλ (qt = i qt+1 = j|o). If there is more than one sequence of interest I use a
superscript to denote the sequence.
2 EM training with Discrete Observation Models
In this section we review two methods for training standard HMM models with discrete
observations: E-M training and Viterbi training.
2.1 The E-M auxiliary function
Let λ represent the current model and λ̄ represent a candidate model. Our objective is to
make Pλ̄ (o) ≥ Pλ (o), or equivalently log Pλ̄ (o) ≥ log Pλ (o).
P
Due to the presence of stochastic constraints (e.g., aij ≥ and j aij = 1) it turns out to
be easier to maximize an auxiliary function Q(·) rather than to directly maximize log P λ̄ .
The E-M auxiliary function is defined as follows:
Q(λ, λ̄) =
X
Pλ (q|o)log Pλ̄ (qo)
(1)
q
Here we show that Q(λ, λ̄) ≥ Q(λ, λ) → log Pλ̄ (o) ≥ log Pλ (o).
For any model λ or λ̄ it must be true that
Pλ̄ (o) =
Pλ̄ (oq)
Pλ̄ (q|o)
(2)
or logPλ̄ (o) = logPλ̄ (oq) − logPλ̄ (q|o).
Also,
logPλ̄ (o) =
X
Pλ (q|o)logPλ̄ (o)
(3)
q
since logPλ̄ (o) is a constant. Thus, from equation 2 it follows that
log Pλ̄ (o) =
X
Pλ (q|o)logPλ̄ (oq) −
q
= Q(λ, λ̄) −
X
Pλ (q|o)logPλ̄ (q|o)
(4)
q
X
Pλ (q|o)logPλ̄ (q|o)
(5)
q
Applying the Q(·, ·) function to (λ, λ̄) and to (λ, λ), it follows that,
Q(λ, λ̄) − Q(λ, λ) = log Pλ̄ (o) − log Pλ (o) − KL(λ, λ̄)
(6)
where KL(·, ·) is the Kullback-Leibler criterion (relative entropy) of the probability distribution Pλ (q|o) with respect to the probability distribution Pλ̄ (q|o)
KL(λ, λ̄) =
X
Pλ (q|o)log
q
Pλ (q|o)
Pλ̄ (q|o)
(7)
Rearranging terms,
log Pλ̄ (o) − log Pλ (o) = Q(λ, λ̄) − Q(λ, λ) + KL(λ, λ̄)
(8)
and since the KL criterion is always positive, it follows that if
Q(λ, λ̄) − Q(λ, λ) ≥ 0
(9)
log Pλ̄ (o)log Pλ (o) ≥ 0
(10)
then
2.2 The overall training sample E-M function
We defined the overall E-M function
Q(λ, λ̄) =
X
Pλ (q|o)log Pλ̄ (qo)
(11)
q
with o including the entire set of sequences in the training sample. Assuming that the
sequences are independent, it follows that
Q(λ, λ̄) =
X
Pλ (q|o)
q
X
log Pλ̄ (q l ol ) =
l
XX
l
(log Pλ̄ (q l ol ))Pλ (q|o)
(12)
q
And since each state sequence q l depends only on the corresponding observation sequence
ol it follows that
Q(λ, λ̄) =
XX X
l
(log Pλ̄ (q l ol ))
K
Y
Pλ (q m |o)
(13)
m=1
q l q−q l
where q − q l = (q 1 , ..., q l−1 , q l+1 , ..., q K ) represents an entire collection of sequences
except for the sequence q l . Thus
Q(λ, λ̄) =
XX X
l
=
XX
l
(log Pλ̄ (q l ol ))Pλ (q l |ol )
(log Pλ̄ (q l ol ))Pλ (q l |ol )
X Y
q−q l
and since
K
X Y
Pλ (q m |o) =
(14)
m6=l
q l q−q l
ql
K
Y
Pλ (q m |o)
(15)
m6=l
Pλ (q m |o) = 1
(16)
q−q l m6=l
it follows that the E-M function can be decomposed into additive E-M functions, one per
observation sequence in the training sample:
XX
X
Q(λ, λ̄) =
Pλ (q l |ol ) log Pλ̄ (q l ol ) =
Ql (λ, λ̄)
(17)
l
ql
l
2.3 Maximizing the E-M function
For simplicity let us start with the case in which there is a single sequence. The results
easily generalize to multiple sequences. Since we work with a single sequence we may
drop the l superscript.
Q(λ, λ̄) =
X
Pλ (q|o)log Pλ̄ (qo)
(18)
q
And since,
Pλ̄ (qo) = π̄q1 b̄q1 (o1 )āq1 q2 b̄q2 (o2 )āq2 q3 ...
(19)
it follows that,
Q(λ, λ̄) =
X
Pλ (q|o)log π̄q1 +
(20)
Pλ (q|o)log b̄qt (ot )+
(21)
q
+
Tl X
X
t=1
+
q
TX
l −1 X
t=1
Pλ (q|o)log āqt qt+1
(22)
q
The first term can be expressed as follows
X
X
X
Pλ (q|o)log π̄q1 =
log π̄j
Pλ (q|o)δ(j, q1 )
q
(23)
q
j
where δ(j, q1 ) tells us to include only those cases in which q1 = j. Therefore,
X
Pλ (q|o)δ(j, q1 ) = Pλ (q1 = j|o) = γ1 (j)
(24)
q
The second term can be expressed as follows
Tl X
X
t=1
Pλ (q|o)log b̄qt (ot ) =
q
Tl X X
X
t=1
i
log b̄i (j)
X
Pλ (q|o)δ(i, qt )δ(j, ot )
(25)
q
j
where δ(i, qt )δ(j, ot ) tells us to include only those cases for which qt = i and ot = j.
Therefore,
X
Pλ (q|o)δ(i, qt )δ(j, ot ) = Pλ (qt = i|o)δ(ot , j) = γt (i)δ(ot , j) =
(26)
q
The third term can be expressed as follows
TX
l −1 X
t=1
q
Pλ (q|o)log āqt qt+1 =
TX
l −1 X X
t=1
i
j
log āij
X
Pλ (q|o)δ(i, qt )δ(j, qt+1 )
(27)
q
where δ(i, qt )δ(j, qt+1 ) tells us to include only those cases for which qt = i and qt+1 = j.
Therefore,
X
Pλ (q|o)δ(i, qt )δ(j, qt+1 ) = Pλ (qt = i qt+1 = j|o) = ξt (i, j)
(28)
q
Putting it together
Q(λ, λ̄) =
X
γ1 (j) log π̄j +
(29)
Tl
X
(30)
j
+
XX
i
log b̄i (j) (
+
XX
i
γt (i)δ(ot , j))+
t=1
j
log āij (
TX
l −1
ξt (i, j))
(31)
t=1
j
When there is more than a sequence, we just need to add up over sequences to obtain the
overall Q(·, ·) function
Q(λ, λ̄) =
X
log π̄j (
j
+
XX
i
+
i
Tl
XX
(32)
log āij (
j
γtl (i)δ(olt , j))+
(33)
t=1
l
XX
γ1l (j))+
l
log b̄i (j) (
j
X
l −1
X TX
l
ξtl (i, j))
(34)
t=1
Note that the part of the overall Q(λ, λ̄) function dependent on π̄j is of the form wj log xj
with xj = π̄j and
K
X
wj =
γ1l (j)
(35)
l=1
with constraints
P
j
xj = 1, and xj ≥ 0.
Equivalently, the part of the overall Q(λ, λ̄) function dependent of b̄i (j) is of the form
wj log xj with xj = b̄i (j) and
wj =
Tl
K X
X
γtl (j)δ(olt , j)
(36)
l=1 t=1
with constraints
P
j
xj = 1, and xj ≥ 0.
Finally, the part of the overall Q(λ, λ̄) function dependent of āij is also of the form
wj log xj with xj = āij and
K TX
l −1
X
wj =
ξtl (i, j)
(37)
l=1 t=1
with constraints
P
j
xj = 1, and xj ≥ 0.
It is easy to show
Pthat the maximum of a function of the form wj log xj with constraints
that xj ≥ 0 and j xj = 1 is achieved for
wj
xj = P
j wj
(38)
The parameters of the new model λ̄ that maximize the overall Q(λ, λ̄) function easily
follow:
π̄i
=
b̄i (j)
=
āij
=
PK
γ1 (i)
K
PK
PTl
l=1
l=1
γt (i)δ(olt ,j)
P Tl
l
t=1 γt (i)
l=1
t=1
PK
(39)
PTl −1
t=1 ξt (i,j)
Pl=1
PTl −1 l
K
l=1
t=1 γt (i)
PK
2.4 Obtaining the E-M parameters from the scaled forward and backward
algorithms
Section under construction
2.5 MM Training
MM training (Maximization Maximization) may be seen as an approximation to EM training. In MM training we basically substitute the Expected value operation by a Max operation thus the MM name. Other names for these algorithms are: Viterbi training (because
we use the Viterbi algorithm to do the Max operation) and segmented K-means (K-means
is a classical clustering method that belongs to the MM family).
In Viterbi-based decoding, the degree of match between a model λ and an observation
sequence ol is defined as ρ(λ, ol ) = maxql log Pλ (q l ol ) = log Pλ (q̂ l ol ). We
have seen how for a fixed model λ, the Viterbi recurrence can be used to find q̂ l and
ρ(λ, ol ). When there is more than one training sequences they are assumed independent
P
l
and ρ(λ, o) = K
l=1 ρ(λ, o ). Thus, the optimal states can be found by applying Viterbi
decoding independently for each of the training sequences.
In MM training the objective is to find an optimal point of log Pλ (q l ol ) with the model λ
and the state sequence q as optimizing variables.
To begin with, assume that Viterbi decoding has found the best sequence of hidden states
for the current λ model: q̂ = (q̂ 1 , ..., q̂ K ). Once we have q̂ we find a new model λ̄ such
that log Pλ̄ (o q̂) ≥ log Pλ (o q̂). To do so simply define a dummy model λ̂ such that
Pλ̂ (qo) = δ(q̂, o), thus the dummy model is such that only state sequence q̂ can co-occur
with observation sequence o.
For such model, Pλ̂ (q1 = j|ol ) = δ(i, q̂ l ), Pλ̂ (qt = i|ol ) = δ(i, q̂t ), and Pλ̂ (qt = i qt+1 =
l
j|ol ) = δ(i, q̂t )δ(j, q̂t+1
). Also note that Q(λ̂, λ̄) = log Pλ̄ (o q̂), the function we want
to optimize. Thus, substituting the λ̂ parameters in the standard E-M formulas guarantees
that Q(λ̂, λ̄) ≥ Q(λ̂, λ) or log Pλ̄ (o q̂) ≥ log Pλ (o q̂)
Thus, the MM training rules are as follows:
π̄i
=
b̄i (j)
=
āij
=
PK
δ(i,q̂1l )
K
l=1
PK
l=1
P Tl
δ(i,q̂tl )δ(j,olt )
P Tl
l
t=1 δ(i,q̂t )
l=1
t=1
PK
PK
l=1
(40)
PTl −1
l
)
δ(i,q̂tl )δ(j,q̂t+1
PTl −1
l)
δ(i,q̂
t
t=1
l=1
t=1
PK
Note that Viterbi decoding maximizes log Pλ (q, o) with respect to q for a fixed model (the
first M step) then we maximizes log Pλ (q, o) with respect to λ, for a fixed q (the second
M step). Since we are always maximizing with respect to some variables, log P λ (q o) can
only increase and convergence to a local maximum is guaranteed.
3 Notation for Continuous Density Models
The notation for the continuous case is the same as the discrete case with the following
additional terms.
P Number of dimensions per observation (e.g. cepstral coefficients): o t = (ot1 ...otP ).
M. Number of clusters within a state.
V = {v11 , ..., v1M ..., vN 1 ..., vN M }. Set of possible clusters of external observations (M
clusters per state). Each cluster vij is identified by a state index i and a cluster index j.
mt variable identifying the cluster index of the cluster that occurred at time t. For example
if cluster vik occurred at time t then qt = i and mt = k.
m = (m1 ...mt ). A sequence of cluster indexes, one per time step.
gik = Pλ (mt = k|qt = j). Emission probability (gain) of cluster kvik by state Sj .
φ(·). A kernel function (e.g. Gaussian) to model the probability density of a cluster of
observations produced by a state.
bj (ot ) = Pλ (ot |qt = j).
µik = (µik1 , ..., µikP ). The centroid or prototype of cluster vik .

 2
σik1 σik12 ... σik1p
2
σik2 ... σik2p 
 σ
Σik =  ik21
 The covariance matrix of cluster vik .
...
2
σikp1 σikp2 ... σikp
2
σikn
. The variance of the l th dimension (e.g. cepstral) within cluster vik , a diagonal element of Σik .
σiknm , The covariance between dimension l and dimension m within the cluster v ik an
off-diagonal elements of Σik . Usually assumed zero.
4 Continuous Observation Models
In this section we study the E-M and Viterbi training procedures for continuous observation
HMMs.
4.1 Mixtures of Densities
In the discrete case the observations are discrete, represented by an integer. In the continuous case the observations at each time step are P-dimensional real-valued vectors (e.g.
cepstrals coefficients). In our notation ot = (ot1 , ..., otP ).
The continuous observation model produces sequences of observations in the following
way: At each time step the system generates a hidden state qt according to a a state to state
transition probability distribution aqt−1 qt . Once qt has been generated, the system generates a hidden cluster mt according to a state to cluster emission probability distribution
gqt mt . Once the hidden cluster has been determined, an observation vector is produced
probabilistically according to some kernel probability distribution (e.g. Multivariate Gaussian). We can think of the clusters as low level hidden states embedded within high level
hidden states qt . For example, the high level hidden states may represent phonemes and
the low-level hidden clusters may represent acoustic categories within the same phoneme.
For simplicity each state is assumed to have the same number of clusters (M) but the set of
clusters is different from state to state. Thus, there is a total of NxM clusters, M per state.
We represent cluster k of state Sj as vjk and we use the variable mt to identify the cluster
number within a state at time t. Thus if qt = i and mt = k it means that at time t cluster
vjk occurred.
If we know the cluster at time t, the probability density of a vector of continuous observations (e.g. cepstral coefficients) is modeled by a kernel function, usually a multivariate
Gaussian
Pλ (ot |vjk ) = Pλ (ot |qt = j mt = k) = φ(ot , µjk , Σjk )
(41)
Where φ(·) is the kernel function (e.g. multivariate Gaussian) µjk = (µjk1 , ..., µjkP ) is
a centroid or prototype that determines the position of the cluster in P-dimensional space,
and
 2

σjk1 σjk12 ... σjk1P
 σjk21 σ 2
... σjk2P 
jk2

Σjk = 
(42)
 ...

2
σjkP 1 σjkP 2 ... σjkP
is a covariance matrix that determines the width and tilt of the cluster in P-dimensional
2
2
space. The diagonal terms {σjk1
...σjkP
} are the cluster variances for each dimension.
They determine the spread of the cluster on each dimension. The off-diagonal elements are
known as the cluster covariances and they determine the tilt of the cluster. For the Gaussian
case, the kernel function is
φ(ot , µjk , Σjk ) =
1
e−d(ot ,µjk )
(2π)P/2 |Σjk |1/2
(43)
where |Σjk | is the determinant of the variance matrix, and d(ot , µjk ) is known as the Mahalanobis distance between the observation and the kernel’s centroid.
1
0
d(ot , µjk ) = (ot − µjk )Σ−1
(44)
jk (ot − µjk )
2
Since the covariance matrices Σjk are symmetric, each kernel is defined by P (P2+3) parameters (P for the centroids and P (P + 1)/2 for the variances). In practice the off-diagonal
variances are assumed zero, reducing the number of parameters per kernel to 2P. In such
QP
2
case the determinant |Σjk | is just the product of P scalar variances |Σjk | = l=1 σjkl
, and
the Mahalanobis distance becomes a scaled Euclidean distance:
P
d(ot , µjk ) =
1 X (otl − µjkl )2
2
2
σjkl
(45)
l=1
Given a state Sj , the system randomly chooses one of its M possible clusters with state to
cluster emission probability P (mt = k|qt = j). This probability is assumed independent
of t and thus it can be represented by a parameter with no time index. In our notation
P (mt = k|qt = j) = gjk , the gain of the k th cluster embedded in state Sj .
Thus, the overall probability density of the observations generated by a state S j is given by
a weighted mixture of kernel functions.
Pλ (ot |qt = j) =
M
X
P (mt = l|qt = j)P (ot |qt = j, mt = k) =
(46)
l=1
or
bj (ot ) =
M
X
k=1
gjk φ(ot , µjk , Σjk )
(47)
4.2 Forward and backward variables
The un-scaled and the scaled algorithms work the same as in the discrete case. Only now
the emission probability terms bj (ot ) are modeled by a mixture of densities.
M
X
bj (ot ) =
gjk φ(ot , µjk , Σjk )
(48)
k=1
4.3 EM Training
In the continuous case the clusters are low level hidden states mt embedded within high
level hidden states qt . Thus, the E-M function is defined over all possible q m sequences
of high-level and low-level hidden states
Q(λ, λ̄) =
XX
q
Pλ (qm|o)log Pλ̄ (qmo)
(49)
m
and since
Pλ̄ (qmo) = π̄q1 ḡq1 m1 φ(o1 , µ̄q1 m1 , Σ̄q1 m1 ), ..., āqT −1 qT ḡqT mT φ(oT , µ̄qT mT , Σ̄qT mT )
(50)
it follows that
Q(λ, λ̄) =
XX
q
T
−1 X X
X
q
t=1
+
+
T XX
X
q
t=1
Pλ (qm|o)log āqt qt+1 +
(52)
Pλ (qm|o)log ḡqt mt
(53)
Pλ (qm|o)log φ(ot , µ̄qt mt , Σ̄qt mt )
(54)
q
m
m
Since the factors in the first two terms are independent of m they simplify into
XX
X
Pλ (qm|o)log π̄q1 =
Pλ (q|o)log π̄q1
q
(51)
m
T XX
X
t=1
Pλ (qm|o)log π̄q1 +
m
m
(55)
q
and
T
−1 X X
X
t=1
q
Pλ (qm|o)log āqt qt+1 =
m
T
−1 X
X
t=1
Pλ (q|o)log āqt qt+1
(56)
q
These terms are identical as in the discrete case and thus the same training rules for initial
state probabilities and for state transition probabilities apply here.
To find the training formulas for the cluster gains, we focus on the part of Q(·) dependent
on the gain terms. This part can be transformed as follows
T XX
X
t=1
q
m
Pλ (qm|o)log ḡqt (mt ) =
(57)
=
T X
N X
M XX
X
q
t=1 i=1 k=1
=
Pλ (qm|o)log ḡik δ(i, qt )δ(j, mt )
(58)
m
T X
N X
M
X
Pλ (qt = i mt = k|o)log ḡik
(59)
t=1 i=1 k=1
Thus the part of Q(·) that depends on ḡik is of the form wj log xj with xj = ḡik and
wj =
T
X
Pλ (qt = i mt = k|o)
(60)
t=1
P
xj = 1 and xj ≤ 0, with maximum achieved for
wj
xj = P
j wj
(61)
PT
PT
Pλ (qt = i mt = k|o)
t=1 Pλ (qt = i mt = k|o)
=
ḡik = PT t=1
PM
PT
k=1 Pλ (qt = i mt = k|o)
t=1
t=1 Pλ (qt = i|o)
(62)
with constraints
j
Thus
To find the learning rules for the centroids and variances we focus on the part of Q(·) that
depends on the cluster centroids and variances, which is given by the following expression:
T XX
X
q
t=1
Pλ (qm|o)log φ(ot , µ̄qt mt , Σ̄qt mt )
(63)
m
This expression can be transformed as follows
T XX
X
t=1
=
q
Pλ (qm|o)log φ(ot , µ̄qt mt , Σ̄qt mt ) =
T X
N X
M XX
X
t=1 i=1 k=1 q
=
(64)
m
Pλ (qm|o)log φ(ot , µ̄ik , Σ̄ik )δ(i, qt )δ(k, mt )
(65)
m
T X
N X
M
X
Pλ (qt = i mt = k|o)log φ(ot , µ̄ik , Σ̄ik )
(66)
t=1 i=1 k=1
Thus, at a maximum,
T
N M
∂ XXX
Pλ (qt = i mt = k|o)log φ(ot , µ̄ik , Σ̄ik ) = 0
∂ µ̄ikn t=1 i=1
(67)
∂log φ(ot , µ̄ik , Σ̄ik )
1
) = 2 (otl − µ̄ikn )
∂ µ̄ikn
σ̄ikl
(68)
k=1
and since
it follows that at a maximum
T
X
Pλ (qt = i mt = k|o)(otl − µ̄ikn ) = 0
t=1
(69)
or
PT
t=1
µ̄ikn =
Pλ (qt = i mt = k|o) otn
PT
t=1 Pλ (qt = i|o)
(70)
2
A similar argument can be made for the diagonal variance σikl
. In this case
1
(otn − µ̄ikn )2
∂log φ(ot , µ̄ik , Σ̄ik )
= − 2 (1 −
)
2
2
∂ σ̄ikn
2σ̄ikn
σ̄ikn
(71)
Thus, at a maximum
T
X
Pλ (qt = i mt = k|o)(1 −
t=1
(otl − µ̄ikn )2
)=0
2
σ̄ikn
(72)
from which the re-estimation formula easily follows:
σ¯2 ikn =
PT
t=1
Pλ (qt = i mt = k|o)(otl − µ̄ikn )2
PT
t=1 Pλ (qt = i|o)
(73)
Training for the mixture gains, mixture centroids, and mixture variances requires the
P (qt = jmt = k|o) terms, for t = 1...T , j = 1..N , k = 1...M . To obtain these
terms note the following:
P (qt = j mt = k|o) = P (qt = j|o)P (mt = k|qt = j o) =
(74)
= P (qt = j|o)P (mt = k|qt = j ot )
(75)
P (ot mt = k|qt = j)
P (ot |qt = j)
(76)
P (mt = k|qt = j)P (ot |qt = jmt = k)
P (ot |qt = j)
(77)
gjk φ(ot , µik , Σik )
bj (ot )
(78)
= P (qt = j|o)
= P (qt = j|o)
Thus
P (qt = jmt = k|o) = P (qt = j|o)
As in the discrete case, the P (qt = j|o) can be obtained through the scaled feed-forward
algorithm.
For the case with multiple training sequences the overall Q(·) decomposes into additive
Ql (·), one per training sequence. As a consequence we have to add in the numerator and
denominator of the training formulas the effects of each training sequence.
Summarizing, the E-M learning rules for the mixture of Gaussian densities case with diagonal covariance matrices are as follows:
π̄i
=
āij
=
ḡik
=
µ̄ikn
=
2
σ̄ikn
=
PK
Pλ (q1 =j|ol )
K
PK
PTl −1
l=1
l=1
Pλ (qt =i qt+1 =j|ol )
PTl −1
l
t=1 Pλ (qt =i|o )
l=1
t=1
PK
P K P Tl
P (q =i mt =k|ol )
l=1
PK t=1
PT λ t
l
l=1
t=1 Pλ (qt =i|o )
(79)
PK
P Tl
l
l
t=1 Pλ (qt =i mt =k|o ) otn
P
T
l)
P
(q
=i|o
t
λ
t=1
PK
P Tl
l=1
l=1
Pλ (qt =i mt =k|ol ) (otn −µ̄ikn )2
P K P Tl
l
t=1 Pλ (qt =i|o )
l=1
t=1
where l = 1, ..., K indexes the training sequence, t = 1, ..., Tl indexes the time within
a training sequence, i = 1, ..., N and j = 1, ..., N index the hidden state, k = 1, ..., M
indexes the cluster embedded within a state, and n = 1, ..., P indexes the dimension of the
continuous vector of observations (e.g. the cepstral coefficient). The P (q t = jmt = k|o),
Pλ (qt = i|ol ) and Pλ (qt = i qt+1 = j|ol ) terms are obtained from the scaled forwardbackward algorithms according to the following formulas:
P (qt = i, mt = k|o)
= P (qt = i|o) gik
bi (ot )
=
φ(ot , µik , Σik )
=
PM
k=1
φ(ot ,µik ,Σik )
bi (ot )
gik φ(ot , µik , Σik )
(2π)P/2
1
Q
P
n=1
σikn
e−d(ot ,µik )
(80)
PP
otn −µikn 2
)
n=1 ( σikn
d(ot , µik )
=
1
2
Pλ (qt = i|ol )
=
α̂l (i)β̂ l (i)
P t l t l
i α̂t (i)β̂t (i)
Pλ (qt = i qt+1 = j|ol )
= α̂t (i)aij β̂t+1 (j)bj (olt+1 )
4.4 Viterbi decoding
We can use Viterbi decoding to find the best possible sequence of high level hidden states
q and low level clusters m. There are two approaches to this problem. One approach
attempts to find simultaneously the best joint sequence of high-level and low-level states.
Thus the goal is to find q̂ m̂ = arg maxqm Pλ (qm|o). The second approach first finds the
best possible sequence of high level states q̂ = arg maxq Pλ (q|o) and once q̂ has been
found, m̂ is defined as m̂ = arg maxm Pλ (m̂|q̂ o). The two approaches do not necessarily
yield the same results. The second approach is the standard in the literature.
For the second, most used version, of Viterbi decoding, the same Viterbi recurrence as in
the discrete case applies but using the continuous version of b i (ot ). Once we have qˆt , the
desired m̂t is simply the cluster within Sq̂t which is closest (in Mahalanobis distance) to
ot .
4.5 Viterbi training
The objective in Viterbi training (also known as segmental k-means) is to find an optimal point (local maximum) of log Pλ (oq̂ m̂) with q and λ being the optimizing variables. It does not matter how q̂ m̂ are found as long as a consistent procedure is used
throughout training. As in the discrete case we define a dummy model λ̂ such that
Pλ̂ (q l ml |o) = δ(q̂ l , q l )δ(m̂l , ml ). For such model, Pλ̂ (q1l = j|ol ) = δ(i, q̂1l ), Pλ̂ (qtl =
l
l
i|ol ) = δ(i, q̂tl ), Pλ̂ (qtl = i qt+1
= j|ol ) = δ(i, q̂tl )δ(j, q̂t+1
), and Pλ̂ (qtl = i mlt = k|ol ) =
l
l
δ(i, q̂t )δ(k, m̂t ).
As in the discrete case note that Q(λ̂, λ̄) = log Pλ̄ (o q̂ m̂), the function we want to optimize. Thus, substituting the λ̂ parameters in the standard E-M formulas guarantees that
Q(λ̂, λ̄) ≥ Q(λ̂, λ), and thus log Pλ̄ (o q̂ m̂) ≥ log Pλ (o q̂ m̂)
Thus, the Viterbi training rules are as follows:
π̄i
=
āij
=
ḡik
=
µ̄ikn
=
σ¯2 ikn
=
PK
δ(i,q̂1l )
K
l=1
PK
l=1
PTl −1
l
δ(i,q̂tl )δ(j,q̂t+1
)
PTl −1
l
t=1 δ(i,q̂t )
l=1
t=1
PK
PK PTl
δ(i,q̂tl )δ(k,m̂lt )
l=1
PK t=1
PT
l
l=1
t=1 δ(i,q̂t )
PK
PTl
δ(i,q̂tl )δ(k,m̂lt ) oltn
t=1
PT
l
t=1 δ(i,q̂t )
PK
PTl
l=1
l=1
(81)
δ(i,q̂tl )δ(k,m̂lt )(otn −µ̄ikn )2
P K P Tl
l
t=1 δ(i,q̂t )
l=1
t=1
Since Viterbi decoding maximizes Pλ (q, m, o) with respect to q and m and Viterbi training maximizes Pλ (q, m, o) with respect to λ, repeatedly applying Viterbi decoding and
Viterbi training, can only make Pλ (q, m, o) increase and convergence to a local maximum
is guaranteed.
5 Factored Sampling Methods for Continuous State Models
Many recognition problems can be framed in terms of infering something about q t the
internal state of a system, based on a sequence of observations o = o 1 · · · ot . These inferences are in many cases based on estimates of p(qt |o1 · · · ot ). When the states are discrete
and countable, these conditonal state probabilities can be obtained using the forwards algorithm. However, the algorithm cannot be used when the states are continuous. In such case,
direct sampling methods are appropriate. Here is an example of how these methods work.
We start with a sensor model: p(ot |qt ) and a Markovian state dynamics model p(qt+1 |qt ).
Our goal is to obtain estimates of p(qt |ot ) for all t.
1. Recursion
Assume we have an estimate p̂(qt |ot ). Our goal is to update that estimate for the
next time step p(qt+1 |ot+1 ).
(a) First we draw a random sample X from p̂(qt |ot ). This sample will implicitely
define a re-estimation of p(qt |ot ) in terms of a mixture of delta functions:
P
p̂(qt |ot ) = N1 N
i=1 δ(qt , xi )
(b) For each observation xi we obtain another random observation yi using the
state dynamics p(qt+1 |qt = xi ). The new sample Y = {y1 · · · yn } implicitely defines our estimates of p(qt+1 |ot ).
PN
p(qt |ot ) = N1 i=1 δ(qt+1 , yi )
(c) We know that
p(o1 · · · ot )
p(qt+1 |o1 · · · ot+1 ) =
p(qt+1 ot+1 |o1 · · · ot ) =
(82)
p(o1 · · · ot+1 )
p(o1 · · · ot )
p(qt+1 |o1 · · · ot )p(ot+1 |qt+1 )
p(o1 · · · ot+1 )
The fraction is a constant K(ot+1 ) independent of qt+1 , we already have an
estimate of p(qt+1 |ot ) so we just need to weight it by p(ot+1 |qt+1 ).
P
p̂(qt+1 |ot+1 ) = K N1 N
i=1 δ(qt+1 , yi )p(ot+1 |qt+1 ) =
N
=
(N )
P
X
1
δ(qt+1 , yi )p(ot+1 |yi )
i p(ot+1 |yi ) i=1
We can now use p̂(qt+1 |ot+1 ) to estimate parameters like the mean or the
variance
R of the distribution. More generally,
ω̂ = dqt+1 Q(p(qt+1 |ot+1 ), qt+1 )
2. Initialization
The initialization step is basically the same as the recursion step only that instead
of using the state transition probabilities we use the initial state probabilities
(a) Obtain a sample of N random states : X= {x1 · · · xN } from the initial state
probability function π(·). These N samples will implicitely work as our
estimate of the initial state probability.
p̂(q1 ) =
N
1 X
δ(q1 , yi )
N i=1
(83)
(b) Weight each observation by the sensor probability p(o1 |q1 = yi ). This defines our initial distribution estimates
p̂(q0 |o1 ) =
1
N
PN
i=1
p(o1 |xi )
δ(q0 , yi )p(o0 |yi )
(84)
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